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Chapter 6: Problem 54
Let \(T\) be a linear transformation from \(R^{2}\) into \(R^{2}\) such that \(T(1,0)=(1,1)\) and \(T(0,1)=(-1,1) .\) Find \(T(1,4)\) and \(T(-2,1)\).
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Prove that if \(A\) is idempotent and \(B\) is similar to \(A,\) then \(B\) is idempotent. \(\left(\operatorname{An} n \times n \text { matrix } A \text { is idempotent if } A=A^{2} .\right)\)
Give a geometric description of the linear transformation defined by the elementary matrix. $$A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$
Let \(A\) be an \(n \times n\) matrix such that \(A^{2}=O .\) Prove that if \(B\) is similar to \(A,\) then \(B^{2}=O\).
Determine whether the linear transformation is invertible. If it is, find its inverse. $$T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(x_{4}, x_{3}, x_{2}, x_{1}\right)$$
Find \(T(\mathbf{v})\) by using (a) the standard matrix and (b) the matrix relative to \(B\) and \(B^{\prime}\). $$\begin{array}{l} T: R^{2} \rightarrow R^{3}, T(x, y)=(x+y, x, y), \mathbf{v}=(5,4) \\ B=\\{(1,-1),(0,1)\\}, B^{\prime}=\\{(1,1,0),(0,1,1),(1,0,1)\\} \end{array}$$
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