91Ó°ÊÓ

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) \((f,-g)\)

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

Find the standard matrix for the linear transformation \(T\). $$T(x, y)=(x+2 y, x-2 y)$$

(a) find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) and \((b)\) show that \(A^{\prime}\) is similar to \(A,\) the standard matrix for \(T\). $$T: R^{2} \rightarrow R^{2}, T(x, y)=(2 x-y, y-x), B^{\prime}=\\{(1,-2),(0,3)\\}$$

In Exercises \(1-8,\) use the function to find (a) the image of \(\mathbf{v}\) and (b) the preimage of \(\mathbf{w}\). $$T\left(v_{1}, v_{2}\right)=\left(v_{1}+v_{2}, v_{1}-v_{2}\right), \mathbf{v}=(3,-4), \mathbf{w}=(3,19)$$

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

(a) find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) and \((b)\) show that \(A^{\prime}\) is similar to \(A,\) the standard matrix for \(T\). $$T: R^{2} \rightarrow R^{2}, T(x, y)=(2 x+y, x-2 y), B^{\prime}=\\{(1,2),(0,4)\\}$$

Find the standard matrix for the linear transformation \(T\). $$T(x, y)=(3 x+2 y, 2 y-x)$$

Let \(T: R^{2} \rightarrow R^{2}\) be a reflection in the \(y\) -axis. Find the image of each vector. (a) (2,5) (b) (-4,-1) (c) \((a, 0)\) (d) \((0, b)\) (e) \((c,-d)\) (f) \((f, g)\)

(a) find the matrix \(A^{\prime}\) for \(T\) relative to the basis \(B^{\prime}\) and \((b)\) show that \(A^{\prime}\) is similar to \(A,\) the standard matrix for \(T\). $$T: R^{2} \rightarrow R^{2}, T(x, y)=(x+y, 4 y), B^{\prime}=\\{(-4,1),(1,-1)\\}$$