#math/linear-algebra **** **Problem 1:** Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Show that with respect to each choice of bases of $V$ and $W$, the matrix of $T$ has at least $\dim \operatorname{range} T$ nonzero entries. Let $v_{1},\ldots,v_{n}$ be a basis for $V$ and $w_{1},\ldots w_{m}$ be a basis for $W$. Denote $r=\text{dim range }T$. $Tv_{i}=a_{1,i}w_{1}+\ldots+a_{m,i}w_{m}$ for $i=1,\ldots, n$. ATC that the matrix of $T$ has $k<r$ nonzero entries. Then at most $k$ $Tv_{i}\neq{}0$. Suppose $v\in V$, then $Tv=T(\alpha_{1} v_{1}+\ldots+\alpha _{n} v_{n})$=\beta _{1} Tu_{1}+\ldots+\beta _{k}Tu_{k}$ where $u_{1},\ldots,u_{k}$ and associated coefficients, are the $v's$ where $Tv_{i}\neq{}0$. Then $\text{dim range T}\leq{}k<r$. Thus it must be that there are at least $\text{dim range }T$ nonzero entries. You can **not** split the v's into null of T and vectors not in null T. The T doesn't care about your basis formation's way of slicing those independent vectors. **** **Problem 2:** Suppose $D \in \mathcal{L}(\mathcal{P}_3(\mathbb{R}), \mathcal{P}_2(\mathbb{R}))$ is the differentiation map defined by $Dp = p'$. Find a basis of $\mathcal{P}_3(\mathbb{R})$ and a basis of $\mathcal{P}_2(\mathbb{R})$ such that the matrix of $D$ with respect to these bases is $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}.$ Chose $(x^{3},x^{2},x,1)$ to be a basis of $\mathcal{P}_{3}(\mathbb{R})$ and $\left( 3x^{2}, 2x,1 \right)$ be a basis of $\mathcal{P}_{2}(\mathbb{R})$. _Compare the exercise above to Example 3.34. The next exercise generalizes the exercise above._ **** **Problem 3:** Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that there exist a basis of $V$ and a basis of $W$ such that with respect to these bases, all entries of $\mathcal{M}(T)$ are $0$ except that the entries in row $j$, column $j$, equal $1$ for $1 \leq j \leq \dim \operatorname{range} T$. Let $u_{1},\ldots,u_{k}$ be a basis for $\text{null }T$. Extend it with $v_{1},\ldots,v_{n}$ to be a basis of $V$. Let $v\in V$. Then $Tv=T(b_{1}u_{1}+\ldots+b_{k}u_{k}+a_{1}v_{1}+\ldots+a_{n}v_{n})=a_{1}Tv_{1}+\ldots+a_{n}Tv_{n}.$ And, by [[3B3 - Fundamental Theorem of Linear Maps|the fundamental theorem of linear maps]], $Tv_{1},\ldots,Tv_{n}$ is a basis for $\text{range }T$. Let $w_{i}\in W$ be the element $Tv_{i}$ for $i=1,\ldots,n$. Extend $w_{1},\ldots,w_{n}$ with $q_{1},\ldots,q_{m}\in W$ to be a basis of $W$. Since $v_{i}\not\in \text{null }T$ then each $Tv_{i}=w_{i}\neq{}0$. By [[3C2 - Matrices as a Linear Map|definition of matrix of linear maps]], $Tv_{i}=A_{1,i}w_{1}+\ldots+A_{n,i}w_{n}+A_{w+1,i}q_{1}+\ldots+A_{n+m,i}q_{m}=A_{i,i}w_{i}=1\cdot w_{i}$. Thus $A_{i,i}=1$ for $i=1,\ldots,n$. **** **Problem 4:** Suppose $v_1, \ldots, v_m$ is a basis of $V$ and $W$ is finite-dimensional. Suppose $T \in \mathcal{L}(V, W)$. Prove that there exists a basis $w_1, \ldots, w_n$ of $W$ such that all the entries in the first column of $\mathcal{M}(T)$ (with respect to the bases $v_1, \ldots, v_m$ and $w_1, \ldots, w_n$) are $0$ except for possibly a $1$ in the first row, first column. _In this exercise, unlike Exercise 3, you are given the basis of $V$ instead of being able to choose a basis of $V$._ **** **Problem 5:** Suppose $w_1, \ldots, w_n$ is a basis of $W$ and $V$ is finite-dimensional. Suppose $T \in \mathcal{L}(V, W)$. Prove that there exists a basis $v_1, \ldots, v_m$ of $V$ such that all the entries in the first row of $\mathcal{M}(T)$ (with respect to the bases $v_1, \ldots, v_m$ and $w_1, \ldots, w_n$) are $0$ except for possibly a $1$ in the first row, first column. _In this exercise, unlike Exercise 3, you are given the basis of $W$ instead of being able to choose a basis of $W$._ **** **Problem 6:** Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $\dim \operatorname{range} T = 1$ if and only if there exist a basis of $V$ and a basis of $W$ such that with respect to these bases, all entries of $\mathcal{M}(T)$ equal $1$. **** **Problem 7:** Verify 3.36. **** **Problem 8:** Verify 3.38. **** **Problem 9:** Prove 3.52. **** **Problem 10:** Suppose $A$ is an $m$-by-$n$ matrix and $C$ is an $n$-by-$p$ matrix. Prove that $(AC)_{j,\cdot} = A_{j,\cdot}\, C$ for $1 \leq j \leq m$. In other words, show that row $j$ of $AC$ equals (row $j$ of $A$) times $C$. **** **Problem 11:** Suppose $a = \begin{pmatrix} a_1 & \cdots & a_n \end{pmatrix}$ is a $1$-by-$n$ matrix and $C$ is an $n$-by-$p$ matrix. Prove that $aC = a_1 C_{1,\cdot} + \cdots + a_n C_{n,\cdot}.$ In other words, show that $aC$ is a linear combination of the rows of $C$, with the scalars that multiply the rows coming from $a$. **** **Problem 12:** Give an example with $2$-by-$2$ matrices to show that matrix multiplication is not commutative. In other words, find $2$-by-$2$ matrices $A$ and $C$ such that $AC \neq CA$. **** **Problem 13:** Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose $A$, $B$, $C$, $D$, $E$, and $F$ are matrices whose sizes are such that $A(B + C)$ and $(D + E)F$ make sense. Prove that $AB + AC$ and $DF + EF$ both make sense and that $A(B + C) = AB + AC$ and $(D + E)F = DF + EF$. **** **Problem 14:** Prove that matrix multiplication is associative. In other words, suppose $A$, $B$, and $C$ are matrices whose sizes are such that $(AB)C$ makes sense. Prove that $A(BC)$ makes sense and that $(AB)C = A(BC)$. **** **Problem 15:** Suppose $A$ is an $n$-by-$n$ matrix and $1 \leq j, k \leq n$. Show that the entry in row $j$, column $k$, of $A^3$ (which is defined to mean $AAA$) is $\sum_{p=1}^{n} \sum_{r=1}^{n} A_{j,p}\, A_{p,r}\, A_{r,k}.$