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# Linear algebra: exercises

These are exercises for Linear algebra: course.

## Contents

## Overview

## Version 1

Chapter 1, Chapter 2, Chapter 3, Chapter 4 (except Section 4.6), Chapter 5 (Sections 5.1 and 5.2), Chapter 6, Chapter 7 (except Sections 7.4, 7.6), Chapter 8 (except Sections 8.4, 8.5).

1. Is $B=\{1,x,x^{2},x^{3},...\}$ a basis of ${\bf C}({\bf R}),$ the space of all continuous functions?

2. (a) Let \[ A=\left[ \begin{array}{lll} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{array} \right] . \] Compute $A^{2},$ $A^{3}.$ (b) Formulate and prove a theorem about $3\times3$ matrices based on the outcome of part (a).

3. Suppose ${\bf C}({\bf R})$ is the vector space of all continuous functions. Let the function $T:{\bf C}({\bf R})\rightarrow {\bf R}$ be defined by \[ T(f)=f^{\prime}(0),\text{ for all }f\in {\bf C}({\bf R}). \] Show that $T$ is linear.

- Find scalars $a$ and $b$ such that $a(1,2)+b(-1,3)=(1,12)$.
- Let $U$ and $V$ be 2-dimensional subspaces of $R^3$. Prove that $U \cap V \neq 0$.
- Suppose $V$ is a vector space with operations: $v+w=0$ and $rv=0$ for all $v,w∈V,r∈R$. How many elements does $V$ have? Prove by using only the axioms.
- Express $f(x)=(x-1)²-x$ as a linear combination of the power function: $1,x,x²,x³,...$.
- Is it possible that a system of linear equations has (a) no solutions, (b) one solution, (c) two solutions, (3) infinitely many solutions? Give an example or explain why it's not possible.
- Find the set of all vectors in $R^2$ that are orthogonal to $(-1,3)$. Write the set in the standard form of a line through the origin.
- Find the standard inner product of $f(x)=\cos x$ and $g(x)=1$ in $C[0,π]$.
- Suppose $a$ is an element of an inner product space $V$. Let $S$ be the set of all vectors orthogonal to $a$ plus $0.$ Show that $S$ is a subspace of $V$.
- Suppose S is a subspace of $V$ and $\dim S=\dim V$. From the definition of the dimension, prove that $S=V$.
- Suppose $V$ is a subspace of $C^1 [0,1]$ spanned by $\sin x, \cos x$. Define $A$ as $A(f)=f'$. (a) Show that $A$ is well defined on $V$. (b) Find the matrix of $A$.

## Version 2

Basic Linear Algebra by Blyth and Robertson.

- Find four $2×2$ matrices $A$ such that $AA=I$.
- Determine all $2×2$ matrices with $AA=I$.
- Find all $2×2$ matrices $A$ that commute with all $2×2$ matrices.
- Show that the system $ax+by=r, cx+dy=s$ has a unique solution if $ad-bc$ is not zero.
- Show that $[2,4,2],[3,2,0],[1,-2,2]$ are linearly independent.
- Suppose $u$ and $v$ are linearly independent. Let $x=u+v$ and $y=u-v$. Are $x$ and $y$ linearly independent?
- Find the Hermite form of the given $3×4$ matrix.
- Show that row operations can be undone by other row operations.
- Are $[1,2],[1,3],[1,4]$ linearly independent? Prove.
- Find all possible values for $rank A$ if $A$ is an $n×m$ matrix.
- Suppose $a, b, c$ are $3×1$ linearly independent columns. Suppose $A$ is $3×3$ matrix of rank $3$. Are the $3×1$ columns $Aa, Ab, Ac$ linearly independent?
- Find the row rank of a given $4×4$ matrix.
- Suppose $A$ is an $n×n$ matrix with only 0 entries on the diagonal and below. Show that $A^n=0$.
- Solve a given $4×4$ system.
- Suppose $A$ is an $n×n$ matrix with only 0 entries on the diagonal and below. Show $B=I-A$ is invertible and $B^{-1}=I+A+A^2+…+A^{n-1}$. Prove the problem by means of inverses.
- Find the inverse of a given $3×3$ matrix.
- What is the smallest subset of $R$ that contains $1/2$ and closed under (a) addition, (b) multiplication.
- Determine whether $S=\{f \in C^1(R):2f'(x)+x^2f(x)=0$ for all $x\}$ is a subspace of $C^1(R)$.
- Let P be the space of all polynomials. Show that P is not spanned by any finite set.
- Show that $\{ 1,(x-1),(x-1)^2,…\} $ is a basis of $P.$
- Does the set $\{ (1,2,-1),(2,4,2),(1,2,3),(-1,-2,1)\} $ span $R^3$?
- Suppose $S$ is a subspace of $V$ and $\dim S=\dim V$. From the definition of the dimension, prove that $S=V$.
- Find the dimension of the space of all symmetric $3×3$ matrices.
- The scalar multiplication in a vector space is a linear function in some sense. In what sense and why?
- Suppose $f:V\rightarrow W$ is linear and surjective. Suppose that $Span S=V$, where $S$ is a subset of $V$. Show that $Span f(S)=W$.
- Consider the function $f:R^3\rightarrow R^3$ that rotates each point about the $x$-axis through an angle a. Prove that $f$ is linear and find its matrix.
- Prove that $G(f)=f(0)+f'(0)$ is linear.
- Find the determinant of the $7×7$ matrix with the following entries: $1,2,3,…,49$.

## Extra

Suppose you have 110 marbles, 11 marbles of each of 10 different colors. You place some of them in 11 boxes so that they are all different in each. Prove that you can pick a few boxes from these so that the total number of marbles of each color is even.