Additional Exercises to 2nd Edition of Axler

The notation used will not be exactly the same as Axler. For instance, we will denote the set $\{a_1, \ldots, a_n\}$ as $\{a_i\}$ out of laziness.

Preliminaries

Axler does many things right. It jumps straight into the main protagonists of linear algebra — vector spaces — without boring a reader with the details of a field or a vector space. This is all sensible for a first read, but it is useful to eventually learn what a field and vector space actually are.

Field axioms

Definition. A field is a set of elements $F$ , including two special distinct elements $0$ and $1$ , with two binary operators $+$ and $\cdot$ such that

Commutativity:

$a + b = b + a$

$a\cdot b = b\cdot a$

Associativity:

$(a + b) + c = a + (b + c)$

$(a\cdot b) \cdot c = a \cdot (b \cdot c)$

Distributivity: $a\cdot (b + c) = a \cdot b + b \cdot c$

Identity:

$a + 0 = a$

$a \cdot 1 = a$

Inverse

For all $a$ , there exists an element $b$ such that $a + b = 0$ .

For all $a \neq 0$ , there exists an element $b$ such that $a \cdot b = 1$ .

for all elements $a$ , $b$ , $c \in F$ .

Even though the field is really the triple $(F, +, \cdot)$ , we will simply refer to the field as $F$ for brevity.

Typically we omit $\cdot$ when it is clear, e.g. we will write $a\cdot b$ as $ab$ .

Examples of fields include $\mathbb{Q}$ and $\mathbb{R}$ , as well as the integers modulo $p$ , i.e. $\mathbb{F}_p$ . Note that $\mathbb{Z}$ is not a field.

A couple of results you should try to prove:

Prove that if a+b=a+ca + b = a + c, then b+cb + c.
- Note from above that the additive inverse is unique, i.e. if $a + b = a + c = 0$ , then $b = c$ .
Prove that if ab=acab = ac, then either a=0a = 0 or b=cb = c.
- Note from above that the multiplicative inverse of any non-zero element is unique, i.e. if $ab = ac = 1$ and $a\neq 0$ , then $b = c$ .
Prove that $0a = 0$ .

Vector space axioms

Definition. A vector space consists of a set $V$ of vectors — including a special vector $0$ — over a field $F$ of scalars, and two binary operators $+ \colon V \times V \to V$ and $\cdot \colon F\times V \to V$ which satisfy

Commutativity: $u + v = v + u$

Associativity:

$(u + v) + w = u + (v + w)$

$(a \cdot b) \cdot v = a \cdot (b \cdot v)$

Distributivity:

$(a + b) \cdot v = a \cdot v + b \cdot v$

$a \cdot (u + v) = a \cdot u + a \cdot v$

for all $u$ , $v \in V$ and $a$ , $b \in F$ .

Note that $+$ can either represent the binary operator $+ \colon F \times F \to F$ or $+ \colon V \times V \to V$ . Similarly, $\cdot$ can either represent the binary operator $\cdot \colon F \times F \to F$ or $\cdot \colon F \times V \to V$ . In the interest of conciseness, we will not explicitly differentiate the two. You will have to infer.

Furthermore, we denote the vector space as $V$ for brevity, even though it really also involves a field and two binary operators.

A result you should try to prove:

Prove that $0\cdot v = 0$ .

Notes

Existence of direct complement for infinite vector spaces

Given a finite-dimensional vector space $V$ and a subspace $U$ of $V$ , there exists some subspace $W$ of $V$ such that $U \oplus W = V$ . (We can prove this by explicitly extending a basis of $U$ .) However, what if $V$ is infinite? Then our explicit extension of a basis doesn’t work, because infinite vector spaces cannot have a finite basis.

In fact, the existence of this direct complement is guaranteed not through a proof but through the Axiom of Choice. (This statement being true, in other words, is equivalent to AoC.)

Chapter 2

Suppose that {v1,…,vn}\{v_1, \ldots, v_n\} forms a basis of a vector space VV. Prove that
- replacing any element $v_i$ with $kv_i$ where $k$ is a non-zero scalar
- replacing any element $v_i$ with $v_i+v_j$ where $j\neq i$
creates another basis of VV.
Prove that every vector in a vector space has a unique representation in any basis. More concretely, given $\vec{v}$ in a vector space $V$ and a basis $\{\vec{b_i}\}$ of $V$ , show that there is only one choice of scalars $\{k_i\}$ such that $\vec{v}=\sum k_i\vec{b_i}$ .
Prove that if $U$ is a subspace of $V$ and $\dim U = \dim V$ , then $U = V$ .

Chapter 3

Suppose $T$ is invertible. Show that if $\{v_i\}$ is a basis of $V$ , then so is $\{Tv_i\}$ .
Prove that for any invertible linear map $T$ , $kT$ is also invertible for all non-zero scalars $k$ .
Prove that for any non-invertible linear map $T$ , $kT$ is also non-invertible for all scalars $k$ .
Suppose $T \colon V\to V$ is a linear map. Show that there exists some map $S\colon V \to V$ such that $TST = T$ .

Matrix exercises (feel free to skip)

Show the matrix of that $P^{-1}TP$ with respect to a basis $B$ is the matrix of $T$ with respect to the basis $P$ . (Here “the basis P” means the basis of vectors $\{Pb_1, \ldots, Pb_n\}$ , where $\{b_1, \ldots, b_n\}$ is the basis $B$ .)
Show that the product of two square upper-triangular matrices $A$ and $B$ is an upper-triangular square matrix $C$ . Also, show that the $i$ th entry on the diagonal of $C$ is the product of the $i$ th entry onthe diagonals of $A$ and $B$ .

Chapter 5

Suppose that $\{U_i\}$ is a collection of invariant subspaces of $V$ under a linear map $T$ . Show that $\sum U_i$ is also invariant under $T$ .
(CMU 21341 Final, Spring 2011) Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ . Suppose $S, T\in \mathcal{L}(V, V)$ are such that $ST=TS$ . Let $\lambda \in \mathbb{C}$ be an eigenvalue of $S$ . Show that there exists $\mu \in \mathbb{C}$ and $v\in V$ with $v\neq 0$ , such that both $Sv=\lambda v$ and $Tv = \mu v$ .

Chapter 6

Prove the Extended Triangle Inequality: $||\sum v_i|| \leq \sum ||v_i||$ with equality if and only if all $v_i$ are multiples of each other.
Prove that for any inner product space $V$ over the real or complex numbers, $||v_1+\cdots+v_n||^2 \leq n(||v_1||^2+\cdots+||v_n||^2)$ (where $v_i$ are elements of $V$ ).

Chapter 7

Commentary

Theorem 7.25 states any normal operator can be expressed as a block diagonal matrix with blocks of size $1$ or $2$ (and the size $2$ block matrices are scalar multiples of the rotation matrix).

This statement isn’t terrible (knowing the explicit representation of a linear map is useful, I guess), but there’s a much more natural way to state it. Return to the Spectral Theorem: (normal/self-adjoint) operators in ( $\mathbb{C}$ / $\mathbb{R}$ ) have an orthonormal basis of eigenvectors. A (somewhat contrived) reformulation of the Spectral Theorem is that $T$ can be decomposed into invariant orthogonal subspaces of $\text{dim } 1$ . And obviously every subspace of $\text{dim } 1$ is self-adjoint (and thus normal), so we can say

$T$ can be decomposed into normal invariant orthogonal subspaces of dimension $1$ iff it is (normal/self-adjoint) in ( $\mathbb{C}$ / $\mathbb{R}$ ).

So the equivalent reformulation of 7.25 would be

$T$ can be decomposed into normal invariant orthogonal subspaces of dimension $1$ or $2$ iff it is normal in $\mathbb{R}$ .

(Corollary to 7.6) Show that $\text{null } T = \text{null } T^\star$ if $T$ is normal. Also, show that the converse does not hold.
Show that $\text{null } T = \text{null } \sqrt{T^\star T}$ .
(Generalization of uniqueness of polar decomposition) If $R_1$ and $R_2$ are positive operators such that $||R_1v|| = ||R_2v||$ for all $v$ , show that $R_1=R_2$ .

Chapter 8

Commentary

Let’s restate 8.5 and 8.9 in their full forms, which Axler alludes to later in the chapter.

(Higher-powered 8.5) There exists some non-negative integer $m$ such that $\text{null } T^k \subsetneq \text{null } T^{k+1}$ for $k < m$ and $\text{null } T^k = \text{null } T^{k+1}$ for $k \geq m$ .
(Higher-powered 8.9) There exists some non-negative integer $m$ such that $\text{range } T^k \supsetneq \text{range } T^{k+1}$ for $k < m$ and $\text{range } T^k = \text{range } T^{k+1}$ for $k \geq m$ .

Consider a vector space $V$ and a linear map $T\colon V\to V$ . Given two invariant subspaces $U_1$ , $U_2$ whose intersection consists only of the $0$ vector, show that $\text{char}(U_1 + U_2) = \text{char} U_1 \cdot \text{char} U_2$ and $\text{minpoly}(U_1 + U_2) = \text{lcm}(\text{minpoly} U_1, \text{minpoly} U_2)$ .
Suppose $X$ has eigenvalues $\lambda_1, \ldots, \lambda_n$ with multiplicities $m_1, \ldots, m_n$ . Then show $X^k$ has eigenvalues $\lambda_1^k, \ldots, \lambda_n^k$ with multiplicities $m_1, \ldots, m_n$ for all $k \geq 1$ .