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) \((f,-g)\)

(a) identify the transformation, and (b) graphically represent the transformation for an arbitrary vector in \(R^{2}\). $$T(x, y)=(x+4 y, y)$$

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

Let \(B=\\{(1,1,0),(1,0,1),(0,1,1)\\}\) and \(B^{\prime}=\\{(1,0,0)\) \((0,1,0),(0,0,1)\\}\) be bases for \(R^{3},\) and let \(A=\left[\begin{array}{rrr}\frac{3}{2} & -1 & -\frac{1}{2} \\ -\frac{1}{2} & 2 & \frac{1}{2} \\ \frac{1}{2} & 1 & \frac{5}{2}\end{array}\right]\) be the matrix for \(T: R^{3} \rightarrow R^{3}\) 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 \([\mathbf{v}]_{B^{\prime}}=\left[\begin{array}{lll}1 & 0 & -1\end{array}\right]^{T}\) (c) Find \(P^{-1}\) and \(A^{\prime}\) (the matrix for \(T\) relative to \(B^{\prime}\) ). (d) Find \([T(v)]_{g^{\prime}}\) two ways.

Proof Prove that if \(A\) and \(B\) are similar matrices, then \(|A|=|B|\) Is the converse true?

Proof Prove that if \(A\) is an idempotent matrix and \(B\) is similar to \(A,\) then \(B\) is idempotent. (Recall that an \(n \times n\) matrix \(A\) is idempotent when \(A=A^{2}\).)

let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). $$ \operatorname{rank}(T)=2 $$

Writing Explain why two similar matrices have the same rank.

Finding the Inverse of a Linear Transformation In Exercises \(31-36,\) determine whether the linear transformation is invertible. If it is, find its inverse. $$ T(x, y)=(x+y, x-y) $$

let \(T: R^{3} \rightarrow R^{3}\) be a linear transformation. Find the nullity of \(T\) and give a geometric description of the kernel and range of \(T\). $$ \operatorname{rank}(T)=3 $$