91Ó°ÊÓ

Determine whether the function is a linear transformation.$$T: R^{2} \rightarrow R^{2}, T(x, y)=(x, 1)$$

find the kernel of the linear transformation. $$ T: R^{2} \rightarrow R^{2}, T(x, y)=(x+2 y, y-x) $$

(a) identify the transformation, and (b) graphically represent the transformation for an arbitrary vector in \(R^{2}\). $$T(x, y)=(12 x, y)$$

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

(a) identify the transformation, and (b) graphically represent the transformation for an arbitrary vector in \(R^{2}\). $$T(x, y)=(x, 3 y)$$

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

find the kernel of the linear transformation. $$ T: R^{2} \rightarrow R^{2}, T(x, y)=(x-y, y-x) $$

Determine whether the function is a linear transformation. $$T: R^{2} \rightarrow R^{2}, T(x, y)=\left(x, y^{2}\right)$$

The Standard Matrix for a Linear Transformation In Exercises \(1-6,\) find the standard matrix for the linear transformation \(T\).$$ \begin{aligned} &T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(x_{1}-x_{3}, x_{2}-x_{4}, x_{3}-x_{1}, x_{2}+x_{4}\right)\\\ &\mathbf{v}=(1,2,3,-2) \end{aligned} $$

Determine whether the function is a linear transformation. $$T: R^{3} \rightarrow R^{3}, T(x, y, z)=(x+y, x-y, z)$$