91Ó°ÊÓ

define the linear transformation \(T\) by \(T(x)=A x .\) Find (a) the kernel of \(T\) and (b) the range of \(T\). $$ A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 1 & 2 \end{array}\right] $$

Let \(B=\\{(1,3),(-2,-2)\\}\) and \(B^{\prime}=\\{(-12,0),(-4,4)\\}\) be bases for \(R^{2},\) and let \(A=\left[\begin{array}{ll}3 & 2 \\ 0 & 4\end{array}\right]\) be the matrix for \(T: R^{2} \rightarrow R^{2}\) relative to \(B\) (a) Find the transition matrix \(P\) from \(B^{\prime}\) to \(B\) (b) Use the matrices \(P\) and \(A\) to find \([\mathbf{v}]_{B}\) and \([T(\mathbf{v})]_{B}\) where \([\mathbf{v}]_{B^{\prime}}=\left[\begin{array}{ll}-1 & 2\end{array}\right]^{T}\) (c) Find \(P^{-1}\) and \(A^{\prime}\) (the matrix for \(T\) relative to \(B^{\prime}\) ). (d) Find \([T(v)]_{B}\) two ways.

(a) identify the transformation, and (b) graphically represent the transformation for an arbitrary vector in \(R^{2}\). $$T(x, y)=(x, 5 x+y)$$

define the linear transformation \(T\) by \(T(x)=A x .\) Find (a) the kernel of \(T\) and (b) the range of \(T\). $$ A=\left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 2 & 1 \end{array}\right] $$

Determine whether the function is a linear transformation. $$T: R^{2} \rightarrow R^{3}, T(x, y)=\left(x^{2}, x y, y^{2}\right)$$

Finding the Standard Matrix and the Image In Exercises \(11-22,\) (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 \(R^{2}: T(x, y)=(x,-y)\) \(\mathbf{v}=(4,-1)\)

(a) identify the transformation, and (b) graphically represent the transformation for an arbitrary vector in \(R^{2}\). $$T(x, y)=(x, 9 x+y)$$

define the linear transformation \(T\) by \(T(x)=A x .\) Find (a) the kernel of \(T\) and (b) the range of \(T\). $$ A=\left[\begin{array}{rr} 1 & 3 \\ -1 & -3 \\ 2 & 2 \end{array}\right] $$

Find all fixed points of the linear transformation. Recall that the vector \(\mathbf{v}\) is a fixed pointof \(T\) when \(T(v)=v\). A reflection in the \(y\) -axis

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