91Ó°ÊÓ

Determine whether the function is a linear transformation. $$T: M_{2,2} \rightarrow R, T(A)=a+b+c+d, \text { where } A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$

Determine whether the function is a linear transformation. $$T: M_{3,3} \rightarrow M_{3,3}, T(A)=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right] A$$

Let \(A\) and \(B\) be similar matrices. (a) Prove that \(A^{T}\) and \(B^{T}\) are similar. (b) Prove that if \(A\) is nonsingular, then \(B\) is also nonsingular and \(A^{-1}\) and \(B^{-1}\) are similar. (c) Prove that there exists a matrix \(P\) such that \(B^{k}=P^{-1} A^{k} P\)

In Exercises \(17-34\), (a) find the standard matrix \(A\) for the linear transformation \(T,\) (b) use \(A\) to find the image of the vector \(\mathbf{v},\) and (c) sketch the graph of \(\mathbf{v}\) and its image. \(T\) is the reflection through the origin in \(R^{2}: T(x, y)=(-x,-y)\) \(\mathbf{v}=(3,4)\)

The linear transformation \(T\) is represented by \(T(\mathbf{v})=A \mathbf{v} .\) Find a basis for (a) the kernel of \(T\) and (b) the range of \(T\). $$A=\left[\begin{array}{rrrr} 1 & 2 & -1 & 4 \\ 3 & 1 & 2 & -1 \\ -4 & -3 & -1 & -3 \\ -1 & -2 & 1 & 1 \end{array}\right]$$

Find all fixed points of the linear transformation. The vector \(\mathbf{v}\) is a fixed point of \(T\) if \(T(\mathbf{v})=\mathbf{v}\). A reflection in the line \(y=x\)

In Exercises \(17-34\), (a) find the standard matrix \(A\) for the linear transformation \(T,\) (b) use \(A\) to find the image of the vector \(\mathbf{v},\) and (c) sketch the graph of \(\mathbf{v}\) and its image. \(T\) is the reflection in the line \(y=x\) in \(R^{2}: T(x, y)=(y, x)\) \(\mathbf{v}=(3,4)\).

Find all fixed points of the linear transformation. The vector \(\mathbf{v}\) is a fixed point of \(T\) if \(T(\mathbf{v})=\mathbf{v}\). A reflection in the line \(y=-x\)

The linear transformation \(T\) is represented by \(T(\mathbf{v})=A \mathbf{v} .\) Find a basis for (a) the kernel of \(T\) and (b) the range of \(T\). $$A=\left[\begin{array}{rrrrr} -1 & 3 & 2 & 1 & 4 \\ 2 & 3 & 5 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 \end{array}\right]$$

Determine whether the function is a linear transformation. $$T: M_{3,3} \rightarrow M_{3,3}, T(A)=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right] A$$