91Ó°ÊÓ

Interpret the following linear transformation geometrically: \\[T(\vec{x})=\left[\begin{array}{rr}1 & 1 \\\\-1 & 1\end{array}\right] \vec{x}\\]

The matrix \\[\left[\begin{array}{rr}-0.8 & -0.6 \\\0.6 & -0.8\end{array}\right]\\] represents a rotation. Find the angle of rotation (in radians)

Decide whether the matrices in Exercises 1 through 15 are invertible. If they are, find the inverse. Do the computations with paper and pencil. Show all your work. $$\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{array}\right]$$

Interpret the following linear transformation geometrically: \\[T(\vec{x})=\left[\begin{array}{rr}0 & -1 \\\\-1 & 0\end{array}\right] \vec{x}\\]

Compute matrix products column by column and entry by entry. Interpret matrix multiplication in terms of the underlying linear transformations. Use the rules of matrix algebra. Multiply block matrices. If possible, compute the matrix products using paper and pencil. $$\left[\begin{array}{rr} 1 & 2 \\ 2 & 4 \end{array}\right]\left[\begin{array}{rr} -6 & 8 \\ 3 & -4 \end{array}\right]$$

Prove the following facts: a. The \(2 \times 2\) matrix \\[ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \\] is invertible if and only if \(a d-b c \neq 0 .\) Hint: Consider the cases \(a \neq 0\) and \(a=0\) separately b. If \\[ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \\] is invertible, then \\[ \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right] \\] [The formula in part (b) is worth memorizing.

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 2 \end{array}\right]$$

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{rr} 2 & 3 \\ -3 & 2 \end{array}\right]$$

Find the matrices of the linear transformations from \(\mathbb{R}^{3}\) to \(\mathbb{R}^{3}\) given in Exercises 19 through \(23 .\) Some of these transformations have not been formally defined in the text. Use common sense. You may assume that all these transformations are linear. The orthogonal projection onto the \(x-y\) -plane.

Find all matrices that commute with the given matrix \(A\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right]$$