91Ó°ÊÓ

The linear transformation \(T\) is defined by \(T(\mathbf{x})=A \mathbf{x} .\) Find (a) \(\operatorname{ker}(T),\) (b) nullity \((T)\) (c) range \((T),\) and (d) \(\operatorname{rank}(T)\) $$A=\left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]$$

Determine all \(n \times n\) matrices that are similar to \(I_{n}\).

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 \(y\) -axis in \(R^{2}: T(x, y)=(-x, y)\) \(\mathbf{v}=(2,-3)\).

Determine whether the function is a linear transformation. $$T: M_{2,2} \rightarrow M_{2,2}, T(A)=A^{T}$$

Prove that if \(A\) is idempotent and \(B\) is similar to \(A,\) then \(B\) is idempotent. \(\left(\operatorname{An} n \times n \text { matrix } A \text { is idempotent if } A=A^{2} .\right)\)

The linear transformation \(T\) is defined by \(T(\mathbf{x})=A \mathbf{x} .\) Find (a) \(\operatorname{ker}(T),\) (b) nullity \((T)\) (c) range \((T),\) and (d) \(\operatorname{rank}(T)\) $$A=\left[\begin{array}{rr} 3 & 2 \\ -9 & -6 \end{array}\right]$$

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 \(x\) -axis in \(R^{2}: T(x, y)=(x,-y)\) \(\mathbf{v}=(4,-1)\).

Determine whether the function is a linear transformation. $$T: M_{2,2} \rightarrow M_{2,2}, T(A)=A^{-1}$$

The linear transformation \(T\) is defined by \(T(\mathbf{x})=A \mathbf{x} .\) Find (a) \(\operatorname{ker}(T),\) (b) nullity \((T)\) (c) range \((T),\) and (d) \(\operatorname{rank}(T)\) $$A=\left[\begin{array}{rr} 5 & -3 \\ 1 & 1 \\ 1 & -1 \end{array}\right]$$

Let \(A\) be an \(n \times n\) matrix such that \(A^{2}=O .\) Prove that if \(B\) is similar to \(A,\) then \(B^{2}=O\).