Results in Linear Mathematics (P1)

T.W.Körner

May 19, 2003

Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I have starred certain results which seem to me to go beyond a strict interpretation of the syllabus. However whilst it would not, in my opinion, be fair to set such results as bookwork they could well appear as problems. I should very much appreciate being told of any corrections or possible improvements. This document is written in L^{A}TEXand stored in the file labeled ~twk/1B/V1.tex on emu in (I hope) read permitted form. My e-mail address is twk.

1 Vector Spaces

Convention 1.1 We shall write 𝔽 to mean ℝ or ℂ.

Definition 1.2 We call (V, +, ., 𝔽) a vector space over 𝔽 if, whenever u, v, w ∈ V and λ, μ ∈ 𝔽, then u + v ∈ V, λu ∈ V and

(i) (V, +) is an Abelian group (so in particular u + (v + w) = (u + v) + w, u + v = v + u).

(ii) λ(μu) = (λμ)u.

(iii) (λ + μ)u = λu + μu.

(iv) λ(u + v) = λu + λv.

(v) 1u = u.

Lemma 1.3 (i) The zero \underline{0} of (V, +) satisfies 0u = \underline{0} for all u ∈ V. (ii) The additive inverse −u of u ∈ V satisfies −u = (−1)u.

We call \underline{0} the zero vector and write it as 0. (Our general policy of dropping ‘boldface’ u and ‘underline’ \underline{u} in favour of the simple u will not usually lead to ambiguity but if it does we simply revert to the less simple convention.)

Theorem 1.4 If X is any set then the set 𝔽^{X} of functions f : X → 𝔽 is a vector space if we define ‘vector addition’ and ‘multiplication by a scalar’ by

(f + g)(x) = f(x) + g(x), and (λf)(x) = λf(x)

for all x ∈ X where f, g ∈ 𝔽^{X} and λ ∈ 𝔽.

Definition 1.5 If V is a vector space we say that U ⊆ V is a subspace of V if 0 ∈ U and

(λ, μ ∈ 𝔽, u, v ∈ U) ⇒ λu + μv ∈ U.

Lemma 1.6 If U is a subspace of a vector space V then U is itself a vector space.

It is usually easier to use Theorem 1.4 (or its generalisation Theorem 2.7 below) together with Lemma 1.6 to prove that something is a vector space than to verify the axioms in Definition 1.2.

Example 1.7 The space C([0, 1]) of continuous functions f : [0, 1] → 𝔽, the space P of real polynomials P : ℝ → ℝ, the classical spaces 𝔽^{n} and the set J of n × n real matrices all of whose rows and columns add up to the same number can all be made into vector spaces in a natural way.

Definition 1.8 (i) Vectors e_{1}, e_{2}, …, e_{n} span a vector space E if given any e ∈ E we can find λ_{1}, λ_{2}, …, λ_{n} ∈ 𝔽 such that

e = λ_{1}e_{1} + λ_{2}e_{2} + … + λ_{n}e_{n}.

(ii) Vectors e_{1}, e_{2}, …, e_{n} in a vector space E are linearly independent if the only solution of

0 = λ_{1}e_{1} + λ_{2}e_{2} + … + λ_{n}e_{n}

(with λ_{1}, λ_{2}, …, λ_{n} ∈ 𝔽) is λ_{1} = λ_{2} = … = λ_{n} = 0.

(iii) Vectors e_{1}, e_{2}, …, e_{n} form a (finite) basis of a vector space E if they span E and are linearly independent.

We shall not be interested in ‘infinite bases’ (few people are, in an algebraic context) so we shall write ‘basis’ rather than ‘(finite) basis’ from now on.

Lemma 1.9 Vectors e_{1}, e_{2}, …, e_{n} form a basis of a vector space E if and only if the equation

e = λ_{1}e_{1} + λ_{2}e_{2} + … + λ_{n}e_{n}

(with λ_{1}, λ_{2}, …, λ_{n} ∈ 𝔽) has one and only one solution for each e ∈ E.

Definition 1.10 A vector space E is said to be finite dimensional if it has a finite spanning set.

Lemma 1.11 (i) If vectors e_{1}, e_{2}, …, e_{n} span a vector space E then some sub-collection forms a basis.

(ii) Every finite dimensional vector space has a basis.

The key result in the study of finite dimensional vector spaces is the Steinitz Replacement Lemma.

Theorem 1.12 Let E be a vector space. If (A) e_{1}, e_{2}, …, e_{n} span E, and (B) f_{1}, f_{2}, …, f_{m} are linearly independent in E,

then n ≥ m and (possibly after reordering the e_{j}) f_{1}, f_{2}, …, f_{m}, e_{m+1}, e_{m+2}, …, e_{n} span E.

Corollary 1.13 Every finite dimensional space E has an associated dimension N such that

(i) Every basis of E has N elements.

(ii) Every linearly independent collection of vectors in E has at most N elements.

(iii) Every spanning collection of vectors in E has at least N elements.

Corollary 1.14 (i) Any subspace of a finite dimensional space is finite dimensional (and the dimension of the subspace is no greater than the dimension of the space).

(ii) Any linearly independent collection of vectors in a finite dimensional space can be extended to a basis.

Example 1.15 In Example 1.7 the space C([0, 1]) of continuous functions is infinite dimensional and the space J of n × n magic squares is finite dimensional.

Definition 1.16 If V and W are subspaces of a vector space U we write

V + W = {v + w : v ∈ V, w ∈ W }

.

Lemma 1.17 If V and W are subspaces of a vector space U then V + W and V ∩ W are subspaces of U. Further, if V + W is finite dimensional,

dim(V + W) + dim(V ∩ W) = dim(V) + dim(W).

Definition 1.18 Subspaces E_{1}, , E_{2}, …, E_{m} of a vector space E are said to have E as direct sum if and only if the equation

e = e_{1} + e_{2} + … + e_{m}

(with e_{j} ∈ E_{j} for 1 ≤ j ≤ m) has one and only one solution for each e ∈ E. We then write

E = E_{1} ⊕ E_{2} ⊕ … ⊕ E_{m}.

Lemma 1.19 Subspaces E_{1}, , E_{2}, …, E_{m} of a finite dimensional vector space E have E as direct sum if and only if the combination of bases of E_{1}, , E_{2}, …, E_{m} gives a basis of E.

Lemma 1.20 If V and W are subspaces of a vector space U then V ⊕W = U if and only if V + W = U and V ∩ W = {0}.

The reader is warned that Lemma 1.20 does not generalise in the obvious way to direct sums of more than two subspaces.

Example 1.21 In ℝ^{2} if we set

U = {(x, 0) : x ∈ ℝ}
V = {(0, y) : y ∈ ℝ}
W = {(t, t) : t ∈ ℝ}

then U ∩ V = V ∩ W = W ∩ U = {0} and U + V + W = ℝ^{2} but ℝ^{2} is not the direct sum of U, V and W.

Definition 1.22 If V and W are subspaces of a vector space U and V ⊕W = U then W is called a complementary subspace of V in U.

The reader is warned that this definition is a strong competitor for the title of ‘Definition most frequently mangled by undergraduates’. She is also warned that except in the trivial cases V = U and V = {0} the complementary subspace of V in U IS NOT UNIQUE! In Example 1.21 both U and V are complementary subspaces of W in ℝ^{2}.

Example 1.23 Consider the vector space C(ℝ) of continuous functions f : ℝ → ℝ. Let

E = {f ∈ C(ℝ) : f(x) = f(−x) for all x ∈ ℝ}
F = {f ∈ C(ℝ) : f(x) = −f(−x) for all x ∈ ℝ}
G = {f ∈ C(ℝ) : f(x) = 0 for all x ≤ 0}.

Then both F and G are complements of E in C(ℝ).

The course now contains some remarks on quotient groups. These are starred, but only to prevent the examiners going overboard, the actual ideas are very easy.

The development is parallel to, but easier than the development of quotient groups in course C1. Suppose that U is a subspace of a vector space V. We observe that U is a subgroup of (V, +) the the vector space V considered as an Abelian group under addition. We take over from group theory the idea of a coset

v + U = {v + u : u ∈ U}

and observe that the first part of the proof of Lagrange’s theorem shows that the cosets form a disjoint cover of V.

Lemma 1.24 Let U be a subspace of a vector space V. Then (i) ⋃_{v∈V} (v + U) = V. (ii) If v, w ∈ V then either (v + U) ∩ (w + U) = ∅ or v + V = w + I.

The remarkable thing is that we can define addition and scalar multiplication of these cosets in a natural way. (Of course, we can deal with addition by noting that any subgroup of an Abelian group is normal and quoting course C1 but it is just as easy to do things directly.)

Lemma 1.25 If U is a subspace of a vector space V over 𝔽 and

v_{1} + U = v_{2} + U, w_{1} + U = w_{2} + U, λ ∈ 𝔽

then

(v_{1} + w_{1}) + U = (v_{2} + w_{2}) + U, λv_{1} + U = λv_{2} + U.

Definition 1.26 If U is a subspace of a vector space V over 𝔽 we write V/U for the set of cosets of U and define addition and scalar multiplication on V/U by

(v + U) + (w + U) = (u + w) + U, λ(v + U) = λv + U.

Note that 0(v + U) = 0v + U = U.

Lemma 1.27 If U is a subspace of a vector space V over 𝔽 then V/U with addition and scalar multiplication as in the previous definition is a vector space over 𝔽.

We call V/U a quotient space (or a quotient vector space).

The reader will natural expect us to produce an isomorphism theorem. We shall do so in Theorem 2.26 but first we need to discuss linear maps.

2 Linear Maps

Definition 2.1 If U and V are vector spaces over 𝔽 then the map α : U → V is said to be linear if

α(λ_{1}u_{1} + λ_{2}u_{2}) = λ_{1}α(u_{1}) + λ_{2}α(u_{2})

for all u_{j} ∈ U_{j}, λ_{j} ∈ 𝔽.

If abstract algebraists were the only people to use vector spaces then ‘linear maps’ would be called ‘vector space homomorphisms’. The following definitions may have been mentioned in previous courses.

Definition 2.2 Let f : X → Y.

(i) We say that f is injective if f(x_{1}) = f(x_{2}) implies x_{1} = x_{2}.

(ii) We say that f is surjective if, given y ∈ Y, we can find an x ∈ X such that f(x) = y.

(iii) We say that f is bijective if it is both injective and surjective.

If f : X → Y is bijective then there is a unique function f^{−1} : Y → X (the inverse of f) such that f^{−1}(f(x)) = x for all x ∈ X and f(f^{−1}(y)) = y for all y ∈ Y.

Definition 2.3 If U and V are vector spaces over 𝔽 the linear map α : U → V is said to be an isomorphism if it is a bijection.

Lemma 2.4 If α : U → V is an isomorphism then α^{−1} : V → U is also linear (and so also an isomorphism).

Lemma 2.5 If U and V are vector spaces over 𝔽 and α : U → V is linear then

(i) α(U) = {α(u) : u ∈ U} is a subspace of V.

(ii) α^{−1}(0) = {u ∈ U : α(u) = 0} is a subspace of U.

(iii) α is injective if and only if α^{−1}(0) = {0}.

We call α^{−1}(0) the null space of α and α(U) the range space of α. (Note that, unless α is bijective, there is no inverse function α^{−1}.)

Theorem 2.6 (The Classification Theorem For Finite Dimensional Vector Spaces) Every vector space over 𝔽 of dimension N is isomorphic to 𝔽^{N}.

We now give the promised generalisation of Theorem 1.4 Theorem 2.7 If X is any set and U any vector space over 𝔽 then the set U^{X} of functions f : X → U is a vector space over 𝔽 if we define ‘vector addition’ and ‘multiplication by a scalar’ by

(f + g)(x) = f(x) + g(x), and (λf)(x) = λf(x)

for all x ∈ X where f, g ∈ X^{U} and λ ∈ 𝔽.

Theorem 2.8 If U and V are vector spaces over 𝔽 then the set L(U, V) of linear maps forms a vector space if we define ‘vector addition’ and ‘multiplication by a scalar’ by

(α + β)(u) = α(u) + β(u), and (λα)(u) = λα(u)

for all u ∈ U where α, β ∈ L(U, V) and λ ∈ 𝔽.

The operation of composition interacts with the operation of addition just defined in a suggestive way.

Theorem 2.9 If U, V and W are vector spaces over 𝔽 and α, β ∈ L(U, V), γ, δ ∈ L(V, W), ϵ ∈ L(W, X) then

ϵ(γα) = (ϵγ)α, γ(α + β) = γα + γβ, (γ + δ)α = γα + δα.

Theorems 2.7 and 2.9 apply in particular to L(U, U).

Theorem 2.10 If U is a vector space over 𝔽 then L(U, U) is a vector space over 𝔽 which obeys the additional laws

α(βγ) = (αβ)γ, α(β + γ) = αβ + αγ, (α + β)γ = αγ + βγ, ια = αι,

where ι is the identity map.

(A vector space which obeys these laws is called an ‘algebra’ but the definition is not part of this course.)

Theorem 2.11 If U is a vector space over 𝔽 then the set of bijective (‘invertible’) maps in L(U, U) form a group GL(U) under composition with unit the identity map ι.

GL(U) is called ‘the general linear group’ and its elements are called automorphisms.

Theorem 2.10 forms a link with an older but not unsuccessful tradition of formal symbolic manipulation.

Example 2.12 (i) Consider the vector space C_{ℂ}^{∞}(ℝ) of infinitely differentiable functions f : ℝ → ℂ. The map D : C_{ℂ}^{∞}(ℝ) → C_{ℂ}^{∞}(ℝ) is a well defined linear map. If a_{0}, a_{1}, …, a_{n} ∈ ℝ then, taking D^{0} = I the identity function

(∑_{r=0}^{n} a_{r}D^{r}) f = ∑_{r=0}^{n} a_{r}f^{(r)}.

(ii)^{∗} If a ∈ ℂ then D − aI is surjective with a one dimensional nul space. The map T_{a} : C_{ℂ}^{∞}(ℝ) → C_{ℂ}^{∞}(ℝ) defined by

(T_{a}f)(t) = e^{at} ∫_{0}^{t} f(x)e^{−ax}dx

is linear and injective but not surjective. We have

(D − aI)T_{a} = I.

The general solution f of

(D − aI)f = g

(with f, g ∈ C_{ℂ}^{∞}ℝ) is f = T_{a}g + h with h ∈ (D − aI)^{−1}(0).

(iii)^{∗} If a_{j} ∈ ℂ for 1 ≤ j ≤ n then (D − a_{1}I)(D − a_{2}I) … (D − a_{n}I) is surjective with an n dimensional nul space H. The equation

(D − a_{1}I)(D − a_{2}I) … (D − a_{n}I)f = g

(with f, g ∈ C_{ℂ}^{∞}(ℝ)) always has a solution f_{0} (the ‘particular integral’) and its general solution is f = f_{0} + h where h (the ‘complementary function’) lies in the n dimensional nul space H.

Although general theorems about linear maps are often best viewed geometrically or abstractly, particular computations require matrices.

Theorem 2.13 Suppose that U is a finite dimensional vector space over 𝔽 with basis u_{1}, u_{2}, …, u_{m} and V is a finite dimensional vector space over 𝔽 with basis v_{1}, v_{2}, …, v_{n}. If α : U → V is linear then α has an associated n × m matrix A = (a_{ij}) with entries in 𝔽 given by

α(u_{j}) = ∑_{i=1}^{n} a_{ij}v_{i}. (∗)

Automatically

α(∑_{j=1}^{m} x_{j}u_{j}) = ∑_{i=1}^{n} (∑_{j=1}^{m} a_{ij}x_{j}) v_{i}. (∗∗)

Conversely if A is an n × m matrix with entries in 𝔽 then the formula (∗∗) defines a linear map α : U → V which has A as associated matrix.

We say that A is the matrix of α with respect to the bases u_{1}, u_{2}, …, u_{n} of U and v_{1}, v_{2}, …, v_{n} of V. Equation (∗) is conventional (in the same way that making clocks go clockwise is conventional) but it represents a universal convention which you should follow.

Theorem 2.14 Suppose that U is a finite dimensional vector space over 𝔽 with basis u_{1}, u_{2}, …, u_{m}, V is a finite dimensional vector space over 𝔽 with basis v_{1}, v_{2}, …, v_{n} and W is a finite dimensional vector space over 𝔽 with basis w_{1}, w_{2}, …, w_{p}. If α, β ∈ L(U, V) have matrices A and B with respect to the bases u_{1}, u_{2}, …, u_{m} of U, v_{1}, v_{2}, …, v_{n} of V and γ ∈ L(V, W) has matrix C with respect to the bases v_{1}, v_{2}, …, v_{n} of V and w_{1}, w_{2}, …, w_{p} of W and λ ∈ 𝔽 then α + β has matrix A + B and λα has matrix λA with respect to the bases u_{1}, u_{2}, …, u_{m} of U and v_{1}, v_{2}, …, v_{n} of V and γβ has matrix CB with respect to the bases u_{1}, u_{2}, …, u_{n} of U and w_{1}, w_{2}, …, w_{p}of W.

Here A + B is the matrix E given by

e_{ij} = a_{ij} + b_{ij} for 1 ≤ i ≤ n, 1 ≤ j ≤ m,

λA is the matrix F given by

f_{ij} = λa_{ij} for 1 ≤ i ≤ n, 1 ≤ j ≤ m,

and CB is the matrix G given by

g_{rj} = ∑_{i=1}^{n} c_{ri}b_{ij} for 1 ≤ r ≤ p, 1 ≤ j ≤ m

.

Notice that if, in Theorem 2.13, we write

α (∑_{j=1}^{m} x_{j}u_{j}) = ∑_{i=1}^{n} y_{i}v_{i}

and write x for the column vector (i.e. m × 1 matrix) (x_{j}), y for the column vector (y_{i}) then equation (∗∗) becomes

y = Ax.

The exact correspondence between linear maps of finite dimensional vector spaces and matrices means that we can rewrite theorems about maps as theorems about matrices. For example, restricting Theorem 2.10 to finite dimensional spaces, we obtain the following matricial translation.

Theorem 2.15 If we use the standard matrix addition, multiplication and multiplication by a scalar the set M_{n}(𝔽) of n × n matrices with entries in 𝔽 is a vector space satisfying the further rules

A(B + C) = AB + AC, A(B + C) = AB + AC, (A + B)C = AC + BC, IA = AI = A,

where I is the identity matrix with (i, j)th entry δ_{ij}.

Conversely we can use results about matrices to obtain results on linear maps between finite dimensional spaces.

Theorem 2.16 (i) The vector space M_{nm}(𝔽) of n × m matrices over 𝔽 has dimension nm.

(ii) If U and V are vector spaces of dimension n and m then L(U, V) has dimension nm.

Since Theorem 2.6 tells us that all finite dimensional vector spaces are isomorphic to 𝔽^{n} for some n and since linear maps between such spaces can always be represented by matrices, it is clearly possible to do all questions involving finite dimensional vector spaces by taking bases and using co-ordinates. There are however several reasons for using co-ordinate free methods when possible.

• As Maxwell pointed out, co-ordinate free methods often give a better formulation of the underlying physical or geometric problem.

• For analysts and most physicists the study of finite dimensional is merely a prelude to the study of infinite dimensional spaces where coordinate methods are often not available.

• One of the reasons for doing the course P1 is to learn more abstract modes of thought. Sticking to concrete co-ordinate systems is hardly the way to go about it.

Returning to the ideas associated with Lemma 2.5 we make the following definitions.

Definition 2.17 If U and V are finite dimensional vector spaces over 𝔽 and α : U → V is linear then

(i) The rank r(α) of α is the dimension of the range space α(U).

(ii) The nullity n(α) of α is the dimension of the nul space α^{−1}(0).

Theorem 2.18 If U and V are finite dimensional vector spaces over 𝔽 and α : U → V is linear then

r(α) + n(α) = dim U (∗).

Further we can find bases for U and V such that the matrix A associated with α is given by a_{ij} = 1 if 1 ≤ i = j ≤ r(α), b_{ij} = 0 otherwise.

Formula (∗) is the famous ‘rank-nullity formula’.

Theorem 2.19 If U is a vector space over 𝔽 and α : U → V is linear then

U ⊇ α(U) ⊇ α^{2}(U) ⊇ … ⊇ α^{k}(U) ⊇ α^{k+1}(U) ⊇ ….

If α^{l}(U) = α^{l+1}(U) then α^{k}(U) = α^{l}(U) for all k ≥ l.

If U is finite dimensional then

dim(U) ≥ r(α) ≥ r(α^{2}) ≥ …,

and there exists an l with l ≤ dim(U) such that r(α^{k}) > r(α^{k+1}) for k < l and r(α^{k}) = r(α^{l}) for k ≥ l. Further

r(α^{k}) − r(α^{k+1}) ≥ r(α^{k+1}) − r(α^{k+2})

for all k.

Definition 2.20 If A is a matrix with entries in 𝔽 then the column rank of A is the dimension of the space spanned by the column vectors of A and the row rank of A is the dimension of the space spanned by the row vectors of A.

Lemma 2.21 If A is matrix associated with a linear map α then the column rank of A is the rank of α.

Later, in Theorem 3.17 (and in a ‘more natural manner’ in Theorem 6.10) we shall see that the row and column rank of a matrix are the same so that we can speak just of the ‘rank’ of a matrix. The following theorem used to be a high point of the first course in linear algebra but is now not even in the syllabus.

Theorem 2.22^{∗} Let A be an m × n matrix and b an m × 1 matrix (i.e. a column vector) over 𝔽. Observe that the column rank of A is no greater than m. We define the ‘augmented matrix’ (A|b) to be the m×(n+1) matrix whose first m columns are the columns of A and whose last column is b. Consider the system of m equations

∑_{j=1}^{n} a_{ij}x_{j} = b_{i} [1 ≤ i ≤ m], (∗)

which we write in matrix form as Ax = b (where x is the n × 1 matrix with entries x_{j}).

(i) The system (∗) of equations has a solution if and only if the column ranks of A and the augmented matrix (A|b) are the same.

(ii) If A has column rank m then (∗) always has a solution. If A has column rank less than m (and so, in particular, if n < m) then there exist choices of b for which (∗) has no solution.

(iii) The column rank of A is no greater than n and there exists a subspace V of the space of n × 1 matrices of dimension precisely n−column rank(A) such that, if x′ is a solution of (∗), then x is a solution of (∗) if and only if x − x′ ∈ V. (In particular if m < n (∗) can not have a unique solution.)

(iv) The system (∗) has 0, 1 or infinitely many solutions. It is a necessary (but not a sufficient) condition for (∗) to have a unique solution for each b that n = m.

As we have already noticed, the matrix associated with a linear map depends on the bases chosen.

Theorem 2.23 [The Change of Basis Theorem] Suppose that U is a finite dimensional vector space over 𝔽 with two bases u_{1}, u_{2}, …, u_{m} and u′_{1}, u′_{2}, …, u′_{m} and V is a finite dimensional vector space over 𝔽 with two bases v_{1}, v_{2}, …, v_{n} and v′_{1}, v′_{2}, …, v′_{n}. Then we can find an m×m matrix P and an n×n matrix Q such that

u_{j} = ∑_{i=1}^{m} p_{ij}u′_{j} for 1 ≤ i ≤ n

v_{s} = ∑_{r=1}^{n} q_{rs}u′_{r} for 1 ≤ s ≤ m.

The matrices P and Q are invertible and if α ∈ L(U, V) has matrix A with respect to the bases u_{1}, u_{2}, …, u_{m} of U and v_{1}, v_{2}, …, v_{n} of V and matrix B with respect to the bases u′_{1}, u′_{2}, …, u′_{m} of U and v′_{1}, v′_{2}, …, v′_{n} then

B = Q^{−1}AP.

Corollary 2.24 Let U be an m dimensional vector space over 𝔽 and V an n dimensional vector space. Two m × n matrices A and B can represent the same linear map from U to V (with respect to appropriately chosen bases) if and only if there exist a non singular m × m matrix P and a non singular n × n matrix Q such that B = QAP.

Using Theorem 2.18 we obtain the first and easiest result on ‘canonical forms’ of matrices.

Corollary 2.25 An m × n matrix A over 𝔽 has column rank r if and only if there exist a non singular m × m matrix P and a non singular n × n matrix Q such that B = QAP is an m × n matrix with b_{ij} = 1 if 1 ≤ i = j ≤ r, b_{ij} = 0 otherwise.

The reader will already have seen a computational proof of this based on Gaussian elimination.

We complete this section by proving an isomorphism theorem which directly parallels the isomorphism theorem for groups. It is starred but very easy.

Theorem 2.26 If U and V are vector spaces over 𝔽 and α : U → V is linear then there is a natural isomorphism α̃ : U/α^{−1}(0) → α(U) given by

α̃(u + α^{−1}(0)) = α(u).

As usual, a key point is to show that α̃ is well defined.

3 Endomorphisms

If α ∈ L(U, V) we defined a matrix associated with α in terms of a basis of U and a basis of V. If U = V, i.e. if α is an endomorphism, is seems reasonable only to use one basis.

Definition 3.1 If V is a vector space over 𝔽 with basis v_{1}, v_{2}, …, v_{n}. We say that the linear map α : V → V has the n × n matrix A with respect to the given basis if

α(v_{j}) = ∑_{i=1}^{n} a_{ij}v_{i}.

In this context the change of basis theorem takes the following form.

Theorem 3.2 (The Change of Basis Theorem For Endomorphisms) Suppose that V is a finite dimensional vector space over 𝔽 with two bases v_{1}, v_{2}, …, v_{n} and v′_{1}, v′_{2}, …, v′_{n}. Then we can find an n × n matrix P such that

u_{j} = ∑_{i=1}^{m} p_{ij}u′_{j} for 1 ≤ i ≤ n.

The matrix P is invertible and if the linear map α : V → V has matrix A with respect to the base u_{1}, u_{2}, …, u_{m} and matrix B with respect to the base v′_{1}, v′_{2}, …, v′_{n} then

B = P^{−1}AP.

Corollary 3.3 Let V be an n dimensional vector space over 𝔽. Two n × n matrices A and B can represent the same endomorphism of V (with respect to appropriately chosen bases) if and only if there exists a non singular n × n matrix P such that B = P^{−1}AP.

We are particularly interested in the case when the n×n matrix A can be diagonalised, that is we can find a n×n diagonal matrix D (a matrix D with d_{ij} = 0 for i ≠ j) and an n × n invertible matrix P such that D = P^{−1}AP. In this case we say that A is ‘diagonalisable’. One of the many reasons for being interested in this phenomenon is indicated by the observation that

A^{N} = (P DP^{−1})^{N} = P D^{N}P^{−1}

and that D^{N} is a diagonal matrix whose diagonal entries are the Nth powers of the corresponding diagonal entries of D. In the same way an endomorphism of a finite dimensional vector space V is called diagonalisable if we can find a basis of V with respect to which the matrix associated to α is diagonal. It is important to note that (even over ℂ) NOT ALL SQUARE MATRICES ARE DIAGONALISABLE.

Example 3.4 The matrix

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

is not diagonalisable over 𝔽.

With very little exaggeration it may be said that all putative theorems about matrices should be tested on the matrix just given.

The problem of diagonalisation is closely linked to the existence of eigenvalues and eigenvectors.

Definition 3.5 Let V be a vector space over 𝔽 and let α be an endomorphism of V. If e is a non-zero vector, λ ∈ 𝔽 and

αe = λe

then we say that λ is an eigenvalue and e an eigenvector of α.

Eigenobjects (though associated with infinite rather than finite dimensional spaces) occur throughout modern physics.

Lemma 3.6 An endomorphism of a finite dimensional vector space is diagonalisable if and only if the space has a basis of eigenvectors.

Lemma 3.7 Eigenvectors with distinct eigenvalues are linearly independent. In particular an endomorphism of an n dimensional vector space which has n distinct eigenvalues is diagonalisable.

Our treatment of eigenvectors will use determinants. Recall that the signature function ζ is the unique non trivial homomorphism from the group S(n) of permutations on the set {1, 2, …, n} to the multiplicative group {1, −1}.

Definition 3.8 If A is an n × n matrix over 𝔽 then

det A = ∑_{σ∈S(n)} ζ(σ) ∏_{i=1}^{n} a_{i,σi}.

Lemma 3.9 If A is an n × n matrix with rows a_{1}, a_{2}, …, a_{n} let us write

f(a_{1}, a_{2}, …, a_{n}) = det A.

Then (i) f is linear in each variable. (ii) f is alternating in the sense that

f(a_{σ1}, a_{σ2}, …, a_{σn}) = ζ(σ)f(a_{1}, a_{2}, …, a_{n})

for all σ ∈ S(n).

(iii) If e_{i} is the row vector of length n with jth component δ_{ij} then

f(e_{1}, e_{2}, …, e_{n}) = 1.

Moreover conditions (i), (ii) and (iii) determine f completely.

Theorem 3.10 If A and B are n × n matrices over 𝔽 then det A det B = det(AB).

Corollary 3.11 If A and B are matrices associated with an endomorphism of a finite dimensional vector space (with respect to appropriate bases) then det A = det B.

The result just proved enables us to make the following definition

Definition 3.12 If α is an endomorphism of a finite dimensional vector space V associated to a matrix A with respect to some basis we set det α = det A.

It is a blemish on our presentation that we define a co-ordinate free quantity det α via co-ordinates but, since the quantity only exists (in general) for finite dimensional spaces some reference to bases (though not to matrices) is unavoidable. (If the structure of the course permitted it, I would prefer to define determinants via area.)

Corollary 3.13 If α and β are endomorphisms of a finite dimensional vector space V and if ι is the identity map on V then det α det β = det(αβ) and det ι = 1.

Definition 3.14 If A is an n × n matrix over 𝔽 we write A_{ij} for the determinant of the (n − 1) × (n − 1) matrix formed by removing the ith row and jth column from A. The n × n matrix whose (i, j)th entry is (−1)^{i+j}A_{ji} is called the ‘adjugate matrix’ of A and written as adj(A).

So far as we are concerned the only purpose of introducing the adjugate is to prove the last two parts of the following theorem.

Theorem 3.15 (i) If A is an n × n matrix then

A adj(A) = adj(A)A = (det A)I.

(ii) If A is an n × n matrix then A is invertible if and only if det A ≠ 0. (iii) An endomorphism α of a finite dimensional vector space V over 𝔽

is an automorphism (i.e. an isomorphism of V with itself) if and only if det α ≠ 0.

Before returning, in the next section, to the subject of eigenvalues we look down a couple of by-ways.

Lemma 3.16 Suppose that in Theorem 2.22 we take n = m. Then (∗) has a unique solution for all b if and only if det A ≠ 0.

If A is an n × m matrix with entries a_{ij} the transposed matrix A^{T} is the m × n matrix with entries a_{ji}.

Theorem 3.17 (i) If A is an n × m matrix and B an m × p matrix then (AB)^{T} = B^{T} A^{T}.

(ii) If A is an n × n matrix then det A^{T} = det A.

(iii) If A is an invertible n×n matrix so is A^{T} and then (A^{T})^{−1} = (A^{−1})^{T}. (iv) The row and column rank of a matrix are identical.

In Theorem 6.10 we give a ‘natural’ proof of part (iv) by using the idea of an adjoint map.

4 The Characteristic Polynomial

Definition 4.1 (i) If α is an endomorphism of an n dimensional space over 𝔽 we define its characteristic polynomial P_{α} : 𝔽 → 𝔽 by

P_{α}(t) = det(tι − α).

(ii) If A is an n × n matrix over 𝔽 we define its characteristic polynomial P_{A} : 𝔽 → 𝔽 by

P_{A}(t) = det(tI − A).

Lemma 4.2 If α an endomorphism of a finite dimensional vector space V over 𝔽 is associated to a matrix A with respect to some basis then P_{A} = p_{α}.

Theorem 4.3 Let α be an endomorphism of an n dimensional space V over 𝔽.

(i) P_{α} is a polynomial of degree exactly n.

(ii) λ ∈ 𝔽 is an eigenvalue if and only if P_{α}(λ) = 0.

(iii) It is a necessary (but not, if n ≥ 2, a sufficient) condition for α to be diagonalisable that P_{α} factorises completely into linear factors over 𝔽.

(iv) It is a sufficient (but not, if n ≥ 2, a necessary) condition for α to be diagonalisable that P_{α} has n distinct roots in 𝔽.

(v) If 𝔽 = ℂ then α has at least one eigenvalue (and so has an eigenvector).

The following definition may help illuminate parts (iii) and (iv) of Theorem 4.3.

Definition 4.4 Let α be an endomorphism of a finite dimensional vector space over 𝔽.

(i) We say that an eigenvalue λ has algebraic multiplicity k if (t − λ)^{k} is a factor of P_{α}(t) but (t − λ)^{k+1} is not.

(ii) We say that an eigenvalue λ has geometric multiplicity k if α − λι has nullity k, i.e.

dim{e ∈ V : αe = λe} = k.

Lemma 4.5 The geometric multiplicity of an eigenvalue can not exceed its algebraic multiplicity.

Lemma 4.6 Let α be an endomorphism of an finite dimensional vector space over 𝔽. The following statements about α are equivalent.

(i) α is diagonalisable.

(ii) The sum of the geometric multiplicities of its eigenvalues equals the dimension of V.

(iii) (a) the sum of the algebraic multiplicities of its eigenvalues is the dimension of V and (b) the algebraic multiplicity of each eigenvalue equals its geometric multiplicity.

Although we can not always diagonalise an endomorphism of a finite dimensional vector space over ℂ (see Example 3.4) we can find a basis with respect to which its matrix is triangular.

Theorem 4.7 (i) If α is an endomorphism of an n dimensional vector space V over ℂ then we can find a basis v_{1}, v_{2}, …, v_{n} of V such that

αv_{j} ∈ span{v_{1}, v_{2}, …, v_{j}}.

(ii) If A is an n × n matrix over ℂ we can find an invertible n × n matrix P such that B = P^{−1}AP is (upper) triangular (i.e. b_{ij} = 0 if i > j).

Our method of proof will be induction on the dimension of V starting from the observation that an endomorphism of a complex vector space always has an eigenvector. This method of proof will recur in the course P4. We note a profound difference between the real and complex cases.

Example 4.8 If we work over ℝ the the matrix

A = ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} )

has no eigenvectors and so, in particular we can not find an invertible P such that P^{−1}AP is triangular.

We now turn to the study of the characteristic polynomial for its own sake. By direct calculation, or by using Lemma 4.2 we know that P_{Q^{−1}AQ} = P_{A} whenever Q is invertible. Thus if we write

P_{A}(t) = ∑_{r=0}^{n} c_{r}(A)t^{r}

we know that the coefficients c_{r}(A) are ‘matrix conjugacy class invariants’ in the sense that c_{r}(Q^{−1}AQ) = c_{r}(A). We can readily identify three of these invariants: c_{n}(A) = 1 (which is not very interesting), c_{0}(A) = (−1)^{n} det A (which we already knew to be invariant) and c_{n−1}(A) = − ∑_{i=1}^{n} a_{ii}.

Definition 4.9 If A is an n × n matrix over 𝔽 then we define the trace trA of A by

trA = ∑_{i=1}^{n} a_{ii}.

The trace of an endomorphism on a finite dimensional space is defined to be the trace of any associated matrix.

In the sense made precise by the next lemma, trace is the only linear matrix conjugacy class invariant.

Lemma 4.10 ^{∗} Consider the collection M_{n}(𝔽) of n × n matrices over 𝔽. (i) tr : M_{n}(𝔽) → 𝔽 is a linear map. (ii) If T : M_{n}(𝔽) → 𝔽 is a linear map such that T (P^{−1}AP) = T (A) for

all A ∈ M_{n}(𝔽) and all invertible P ∈ M_{n}(𝔽) then T = (T (I)n^{−1})tr.

If f : ℝ^{n} → ℝ is smooth the Laplacian of f is the trace of its Hessian.

It should be noted that the characteristic polynomial, informative as it is, does not tell us everything about the associated matrix or endomorphism.

Example 4.11 The matrices

( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

have the same characteristic polynomial, yet have different ranks. Further one is diagonalisable and the other is not.

If we write out the characteristic polynomial of an endomorphism of a finite dimensional vector space V as

P_{α}(t) = ∑_{r=0}^{n} c_{r}(α)t^{r}

we see that we may define P_{α}(β) for any endomorphism β of V by

P_{α}(β) = ∑_{r=0}^{n} c_{r}(α)β^{r}.

Observe that P_{α}(β) is itself an endomorphism. If P_{α}(β) is the zero endomorphism we say that β satisfies the characteristic equation of α. In exactly the same way if A and B are n×n matrices over 𝔽 we can define an n×n matrix P_{A}(B). If P_{A}(B) is the zero matrix we say that B satisfies the characteristic equation of A.

Example 4.12 Consider the two 2 × 2 matrices

A = I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , B = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) .

The characteristic polynomial of A is P_{A}(t) = t^{2} − 2t + 1 = (t − 1)^{2} so P_{A}(B) is the non-zero 2 × 2 matrix

( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )

so that B does not satisfy the characteristic equation of A although det(BI − A) is the scalar 0.

Lemma 4.13 (The Cayley-Hamilton Theorem for Triangular Matrices) If A is an n × n triangular matrix over 𝔽 with diagonal entries a_{ii} then

det(tI − A) = (t − a_{11})(t − a_{22}) … (t − a_{nn})

and

(A − a_{11}I)(A − a_{22}) … (A − a_{nn}I) = 0.

At first sight the result just proved seems very special but in Theorem 4.7 we showed that any endomorphism of a finite dimensional vector space over ℂ can be represented (with respect to an appropriate basis) by a triangular matrix. We can thus extend the Cayley-Hamilton Theorem to a wide range of cases.

Corollary 4.14 (i) If V is a finite dimensional vector space over ℂ then any linear map α : V → V satisfies its own characteristic equation.

(ii) Any n × n matrix over ℂ satisfies its own characteristic equation. (iii) Any n × n matrix over ℝ satisfies its own characteristic equation. (iv) If V is a finite dimensional vector space over ℝ then any linear map

α : V → V satisfies its own characteristic equation.

We bring together the results of the corollary as a theorem.

Theorem 4.15 (The Cayley-Hamilton Theorem) (i) Any n × n matrix over 𝔽 satisfies its own characteristic equation.

(ii) If V is a finite dimensional vector space over 𝔽 then any linear map α : V → V satisfies its own characteristic equation.

There are many different proofs of the Cayley-Hamilton Theorem but (so far as I know) no totally trivial ones. If you invent a new proof, first check it against Example 4.12 and then get your supervisor to check it.

5 Jordan Forms

It is unsatisfactory to leave non-diagonalisable endomorphisms (and matrices) over 𝔽 unexamined. In this section we show that they can be fully classified using the Jordan normal form. Although some of the material is more or less explicitly starred some is not and the development is pretty and instructive.

Our first steps are unstarred. Suppose V is a finite dimensional vector space over ℂ and α is an endomorphism of V. The Cayley Hamilton theorem tells us that there is a monic polynomial. P_{α} with P_{α}(α) = 0. It follows that there must be a monic polynomial of least degree which anhilates α.

Definition 5.1 If V is a finite dimensional vector space over ℂ and α is an endomorphism of V the monic polynomial Q of least degree such that Q(α) = 0 is called the minimal polynomial of α.

Lemma 5.2 Suppose that V is a finite dimensional vector space over ℂ and α is an endomorphism of V. The minimal polynomial divides (i.e. is a factor of) every polynomial P with P (α) = 0. In particular the minimal polynomial divides the characteristic polynomial.

Example 5.3 Suppose that V is a finite dimensional vector space over ℂ and α is an endomorphism of V which may be diagonalised with associated diagonal matrix D. If the distinct diagonal entries of D are d_{1}, d_{2}, …, d_{k} then the minimal polynomial Q_{α} of α is given by

Q_{α}(t) = ∏_{j=1}^{k}(t − d_{j}).

In particular the characteristic and minimal polynomials coincide if and only if k = n.

The next results may or may not be starred but are sufficiently important that everyone should know them.

Lemma 5.4 Let P (t) = ∑_{j=0}^{n} a_{j}t^{n} be a polynomial with coefficients in 𝔽 If P (t) = 0 for n + 1 distinct values of t ∈ 𝔽 then a_{0} = a_{1} = ⋯ = a_{n} = 0. In particular P (α) = 0 and P (A) = 0 whenever α is an appropriate endomorphism or A an appropriate matrix.

Theorem 5.5 (Bezout’s Theorem for polynomials) If R_{1}, R_{2}, … R_{k} are polynomials with highest common factor 1 (i.e. if no non-constant polynomial divides all of R_{1}, R_{2}, … R_{k}) then we can find polynomials P_{1}, P_{2}, …, P_{k} such that

∑_{j=1}^{k} P_{j}R_{j} = 1.

We now apply the last two results to the minimal polynomial.

Theorem 5.6 Suppose that V is a finite dimensional vector space over ℂ and α is an endomorphism of V. with minimal polynomial Q. If we write Q(t) = ∏_{j=1}^{k} Q_{j} where Q_{j}(t) = (t − λ_{j})^{n(j)} with n_{j} ≥ 1 and the λ_{j} distinct and set

R_{j}(t) = ∏_{i≠j} Q_{j}

we can find polynomials P_{1}, P_{2}, …, P_{k} such that

∑_{j=1}^{k} P_{j}(α)R_{j}(α) = ι.

Further, if we write V_{j} = R_{j}(α)V the following facts are true.

(i) V_{j} = Q_{j}(α)^{−1}(0).

(ii) α(V_{j}) ⊆ V_{j}.

(iii) V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{k},

Lemma 5.7 With the notation of the preceeding theorem suppose E_{j} is a basis for V_{j}. If α|E_{J} considered as an endomorphism of V_{j} (see (ii) in the preceeding theorem) has matrix A_{J} with respect to E_{j}, then the matrix of α with respect the basis E = ⋃_{j=1}^{k} E_{j} (see (iii) in the preceeding theorem) consists of the square matrices A_{j} along the diagonal with all other entries 0.

Thus we have reduced the problem Problem^{∗} Find a basis for which an endomorphism α has a nice matrix. to a rather simpler problem Problem^{∗∗} Suppose that the endomorphism α of V has the property that (α − λι)^{m} = 0. Find a basis for which an endomorphism α has a nice matrix.

If we observe that whenever α has matrix A with respect to a certain basis then α − λι has matrix A − λI we can make the further reduction to Problem^{∗∗∗} Suppose that the endomorphism α on V has the property that α^{m} = 0. Find a basis for which an endomorphism α has a nice matrix.

This suggests the following definition.

Definition 5.8 An endomorphism α of a vector space is called nilpotent if we can find an m ≥ 0 with α^{m} = 0.

To solve the problems just stated we need the notion of a Jordan matrix. We write J(λ, n) for the n × n matrix with λ’s down the diagonal, 1’s immediately below and zero every where else so that

J(λ, n) = (\begin{matrix}λ & 0 & 0 & ⋯ & 0 & 0 & 0 & 0 \\ 1 & λ & 0 & ⋯ & 0 & 0 & 0 & 0 \\ 0 & 1 & λ & ⋯ & 0 & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & 0 & 1 & λ & 0 \\ 0 & 0 & 0 & ⋯ & 0 & 0 & 1 & λ \end{matrix}) .

We call J(λ, n) a Jordan matrix.

We can now solve Problem^{∗∗∗}.

Theorem 5.9 Suppose that V is a finite dimensional vector space over ℂ and α is a nilpotent endomorphism of V. Then there is basis for V such that α has matrix A (relative to this basis) which consists of zeros except for Jordan matrices J(0, n_{i}) [1 ≤ i ≤ s]. down the diagonal.

The only proofs that I know of Theorem 5.9 are hard and in my view the only reason it is included is to show that your lecturers are cleverer than you are. The proof is starred but not the statement of the theorem. You should, however, convince yourselves that the result is plausible.

It is now easy to solve Problem^{∗∗}.

Lemma 5.10 Suppose that V is a finite dimensional vector space over ℂ and α is an endomorphism of V with the property that (α − λι)^{m} = 0. Then there is basis for V such that α has matrix A (relative to this basis) which consists of zeros except for Jordan matrices J(λ, n_{i}) [1 ≤ i ≤ s]. down the diagonal.

We can now solve our original problem, Problem^{∗}.

Theorem 5.11 (Jordan Normal Form) Suppose that V is a finite dimensional vector space over ℂ and α is an endomorphism of V. Then there is basis for V such that α has matrix A (relative to this basis) which consists of zeros except for Jordan matrices J(λ_{i}, m_{ij}) [1 ≤ m_{ij} ≤ m_{i}, 1 ≤ i ≤ s]. We may demand that the λ_{i} be distinct and that m_{i,1} ≥ mi, 2 ≥ . . ..

A little thought shows that the Jordan form is uniques up to shuffling the diagonal blocks.

The flowing observation is trivial but worth making.

Lemma 5.12 With the notation of Theorem 5.9 α has characteristic polynomial

P_{α}(t) = ∏_{i=1}^{s} ∏_{j=1}^{m_{i}}(t − λ_{i})^{m_{ij}}

and minimal polynomial

P_{α}(t) = ∏_{i=1}^{s}(t − λ_{i})^{m_{i1}}.

The algebraic multiplicity of λ_{i} is ∑_{j=1}^{m_{i}} m_{ij}. and the geometric multiplicity is max_{1≤j≤m_{i}} m_{ij}.

In Course C1 you saw how the Jordan normal form was used to classify the solution of two simultaneous first order differential equations in two unknowns. The extension to n simultaneous first order differential equations in n unknowns is obvious.

6 Dual Spaces

We saw in Theorem 2.8 that the linear maps from one vector space U to another vector space V form a vector space L(U, V). In the two previous sections we investigated the special case when U = V. Now we look at the case when V is one dimensional.

Definition 6.1 If U is a vector space over 𝔽 the vector space of linear maps u′ : U → 𝔽 is called the dual space of U and denoted by U′.

Thus for example the trace map is in the dual space of M_{n}(𝔽).

Example 6.2 Let C^{∞}(ℝ) be the space of infinitely differentiable functions f : ℝ → ℝ. If we fix x ∈ ℝ and set

J(f) = ∫_{−1}^{1} f(t)dt, δ_{x}(f) = f(x), δ′_{x}(f) = f′(x)

then J, δ_{x} and δ′_{x} are all in the dual space of C^{∞}(ℝ).

As the example indicates analysts are very interested in objects which belong to dual spaces whilst algebraists are interested in dual objects in general. However, at this level, we can only say interesting things about the duals of finite dimensional spaces.

Lemma 6.3 Suppose that V is a vector space over 𝔽 with basis e_{1}, e_{2}, …, e_{n}. Then the dual space V′ has the same dimension as V and may be given a basis (the so called dual basis) E_{1}, E_{2}, …, E_{n} with E_{i}(e_{j}) = δ_{ij}.

Lemma 6.4 Let V be a finite dimensional vector space. If we write

Φ(v)(v′) = v′(v)

for all v ∈ V, v′ ∈ V′ then Φ : V → V′′ is an isomorphism.

We already knew from Lemma 6.3 that dim V = dim V′ = dim V′′ but Φ gives us a ‘natural isomorphism’ defined without reference to some particularly chosen basis. The ‘standard convention’ is to write v = Φ(v), i.e. to identify V and V′′ via Φ.

Definition 6.5 If U is a non-empty subset of a vector space V then we define the annihilator U^{∘} of U to be the subset of U given by

U^{∘} = {v′ ∈ V′ : v′(u) = 0 for all u ∈ U}.

Lemma 6.6 If U is a non-empty subset of a vector space V then U is a subspace of V′. If, further, V is finite dimensional and U is a subspace of V then

(i) dim U + dim U^{∘} = dim V.

(ii) Φ(U) = U^{∘∘} so, using the standard convention, U^{∘∘} = U.

We conclude the section and the course by looking at the notion of the adjoint of a linear map.

Lemma 6.7 If U and V are vector spaces over 𝔽 and α ∈ L(U, V) then the equation

α′(v′)(u) = v′(αu)

for all v′ ∈ V, u ∈ U defines an α′ ∈ L(V′, U′). (We call α′ the dual, or the adjoint, of α).

Lemma 6.8 If U and V are finite dimensional vector spaces and α has matrix A with respect to given bases of U and V then α′ has matrix A^{T} with respect to the dual bases.

Lemma 6.9 If U, V and W are vector spaces over 𝔽 and α ∈ L(U, V), β ∈ L(V, W) then (βα)′ = α′β′.

If we choose appropriate bases we recover the matricial formula (BA)^{T} = A^{T} B^{T} for matrices of appropriate sizes.

Theorem 6.10 If U and V are finite dimensional vector spaces. If we adopt the standard convention of identifying U′′ with U and V′′ with V then, if α ∈ L(U, V),

(i) α′′ = α.

(ii) (αU)^{∘} = (α′)^{−1}(0).

(iii) r(α) = r(α′)

(iv) The row rank and the column rank of any matrix are equal.

We have thus fulfilled our promise to prove the equivalence of row and column rank in a natural context.

Lemma 6.11 If α is an endomorphism of a finite dimensional vector space V the α′ is an endomorphism of V′ and det α′ = det α.