91Ó°ÊÓ

a. Prove that if \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is a linear transformation and \(c\) is any scalar, then the function \(c T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) defined by \((c T)(\mathbf{x})=c T(\mathbf{x})\) (i.e., the scalar \(c\) times the vector \(T(\mathbf{x}))\) is also a linear transformation. b. Prove that if \(S: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) and \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) are linear transformations, then the function \(S+T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) defined by \((S+T)(\mathbf{x})=S(\mathbf{x})+T(\mathbf{x})\) is also a linear transformation. c. Prove that if \(S: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}\) and \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) are linear transformations, then the function \(S \circ T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{p}\) is also a linear transformation.

Suppose \(A\) and \(B\) are \(n \times n\) matrices. Prove that if \(A B\) is nonsingular, then both \(A\) and \(B\) are nonsingular. (Hint: First show that \(B\) is nonsingular; then use Theorem \(3.2\) and Proposition 3.4.)

Assume \(A\) and \(B\) are two \(m \times n\) matrices with the same reduced echelon form. Show that there exists an invertible matrix \(E\) so that \(E A=B\). Is the converse true?

Suppose \(A\) is a symmetric \(n \times n\) matrix. If \(\mathbf{x}\) and \(\mathbf{y} \in \mathbb{R}^{n}\) are vectors satisfying the equations \(A \mathbf{x}=2 \mathbf{x}\) and \(A \mathbf{y}=3 \mathbf{y}\), show that \(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal. (Hint: Consider \(A \mathbf{x} \cdot \mathbf{y} .)\)

An \(n \times n\) matrix is called a permutation matrix if it has a single 1 in each row and column and all its remaining entries are 0 . a. Write down all the \(2 \times 2\) permutation matrices. How many are there? b. Write down all the \(3 \times 3\) permutation matrices. How many are there? c. Show that the product of two permutation matrices is again a permutation matrix. Do they commute? d. Prove that every permutation matrix is nonsingular. e. If \(A\) is an \(n \times n\) matrix and \(P\) is an \(n \times n\) permutation matrix, describe the columns of \(A P\) and the rows of \(P A\).

Suppose \(A\) is an invertible matrix and \(A^{-1}\) is known. a. Suppose \(B\) is obtained from \(A\) by switching two columns. How can we find \(B^{-1}\) from \(A^{-1}\) ? (Hint: Since \(A^{-1} A=I\), we know the dot products of the rows of \(A^{-1}\) with the columns of \(A\). So rearranging the columns of \(A\) to make \(B\), we should be able to suitably rearrange the rows of \(A^{-1}\) to make \(B^{-1}\).) b. Suppose \(B\) is obtained from \(A\) by multiplying the \(j^{\text {th }}\) column by a nonzero scalar. How can we find \(B^{-1}\) from \(A^{-1}\) ? c. Suppose \(B\) is obtained from \(A\) by adding a scalar multiple of one column to another. How can we find \(B^{-1}\) from \(A^{-1}\) ? d. Suppose \(B\) is obtained from \(A\) by replacing the \(j^{\text {th }}\) column by a different vector. Assuming \(B\) is still invertible, how can we find \(B^{-1}\) from \(A^{-1}\) ?

(The binomial theorem for matrices) Suppose \(A\) and \(B\) are \(n \times n\) matrices with the property that \(A B=B A\). Prove that for any positive integer \(k\), we have $$ \begin{aligned} (A+B)^{k} &=\sum_{i=0}^{k} \frac{k !}{i !(k-i) !} A^{k-i} B^{i} \\ &=A^{k}+k A^{k-1} B+\frac{k(k-1)}{2} A^{k-2} B^{2}+\frac{k(k-1)(k-2)}{6} A^{k-3} B^{3} \\ &+\cdots+k A B^{k-1}+B^{k} \end{aligned} $$ Show that the result is false when \(A B \neq B A\).

Suppose \(A\) is an \(m \times n\) matrix with a unique right inverse \(B\). Prove that \(m=n\) and that \(A\) is invertible.

We say an \(n \times n\) matrix \(A\) is orthogonal if \(A^{\top} A=I_{n}\). a. Prove that the column vectors \(\mathbf{a}_{1}, \ldots, \mathbf{a}_{n}\) of an orthogonal matrix \(A\) are unit vectors that are orthogonal to one another, i.e., \(\mathbf{a}_{i} \cdot \mathbf{a}_{j}= \begin{cases}1, & i=j \\ 0, & i \neq j\end{cases}\) b. Fill in the missing columns in the following matrices to make them orthogonal: $$ \left[\begin{array}{cc} \frac{\sqrt{3}}{2} & ? \\ -\frac{1}{2} & ? \end{array}\right],\left[\begin{array}{rrr} 1 & 0 & ? \\ 0 & -1 & ? \\ 0 & 0 & ? \end{array}\right], \quad\left[\begin{array}{rrr} \frac{1}{3} & ? & \frac{2}{3} \\ \frac{2}{3} & ? & -\frac{2}{3} \\ \frac{2}{3} & ? & \frac{1}{3} \end{array}\right] $$ c. Show that any \(2 \times 2\) orthogonal matrix \(A\) must be of the form $$ \left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \text { or }\left[\begin{array}{rr} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right] $$ for some real number \(\theta\). (Hint: Use part \(a\), rather than the original definition.) *d. Show that if \(A\) is an orthogonal \(2 \times 2\) matrix, then \(\mu_{A}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is either a rotation or the composition of a rotation and a reflection. e. Prove that the row vectors \(\mathbf{A}_{1}, \ldots, \mathbf{A}_{n}\) of an orthogonal matrix \(A\) are unit vectors that are orthogonal to one another. (Hint: Corollary 3.3.)

a. Show that the only matrix that is both symmetric and skew-symmetric is \(O\). b. Given any square matrix \(A\), show that \(S=\frac{1}{2}\left(A+A^{\top}\right)\) is symmetric and \(K=\frac{1}{2}(A-\) \(A^{\mathrm{T}}\) ) is skew-symmetric. c. Deduce that any square matrix \(A\) can be written in the form \(A=S+K\), where \(S\) is symmetric and \(K\) is skew-symmetric.