91Ó°ÊÓ

find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) $$ \begin{aligned} &T: R^{2} \rightarrow R^{2}, T(x, y)=(2 x-y, y-x)\\\ &B^{\prime}=\\{(1,-2),(0,3)\\} \end{aligned} $$

Let \(T: R^{2} \rightarrow R^{2}\) be a reflection in the \(x\) -axis. Find the image of each vector. (a) \((3,5)\) (b) \((2,-1)\) (c) \((a, 0)\) (d) \((0, b)\) (e) \((-c, d)\) (f) \((f,-g)\)

Use the function to find (a) the image of \(v\) and (b) the preimage of \(\mathbf{w}\). $$\begin{aligned}&T\left(v_{1}, v_{2}\right)=\left(v_{1}+v_{2}, v_{1}-v_{2}\right),\\\&\mathbf{v}=(3,-4), \mathbf{w}=(3,19)\end{aligned}$$

find the kernel of the linear transformation. $$ T: R^{3} \rightarrow R^{3}, T(x, y, z)=(0,0,0) $$

The Standard Matrix for a Linear Transformation In Exercises \(1-6,\) find the standard matrix for the linear transformation \(T\). $$ T(x, y)=(x+2 y, x-2 y) $$

Let \(T: R^{2} \rightarrow R^{2}\) be a reflection in the \(y\) -axis. Find the image of each vector. (a) \((5,2)\) (b) \((-1,-6)\) (c) \((a, 0)\) (d) \((0, b)\) (e) \((c,-d)\) (f) \((f, g)\)

find the kernel of the linear transformation. $$ T: R^{3} \rightarrow R^{3}, T(x, y, z)=(x, 0, z) $$

find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) $$ \begin{aligned} &T: R^{2} \rightarrow R^{2}, T(x, y)=(2 x+y, x-2 y)\\\ &B^{\prime}=\\{(1,2),(0,4)\\} \end{aligned} $$

Use the function to find (a) the image of \(v\) and (b) the preimage of \(\mathbf{w}\). $$\begin{aligned}&T\left(v_{1}, v_{2}\right)=\left(v_{1}, 2 v_{2}-v_{1}, v_{2}\right),\\\&\mathbf{v}=(0,4), \mathbf{w}=(2,4,3)\end{aligned}$$

The Standard Matrix for a Linear Transformation In Exercises \(1-6,\) find the standard matrix for the linear transformation \(T\). $$ T(x, y)=(2 x-3 y, x-y, y-4 x) $$