91Ó°ÊÓ

For the transformation of \(\mathbb{R}^{2}\) with the given matrix, sketch the transform of the square with vertices \((1,1),(2,1),(2,2),\) and (1,2). $$A=\left[\begin{array}{rr}1 & -1 \\\1 & 2\end{array}\right]$$

Find \(\operatorname{Ker}(T)\) and \(\operatorname{Rng}(T),\) and give a geometrical description of each. Also, find \(\operatorname{dim}[\operatorname{Ker}(T)]\) and \(\operatorname{dim} [\operatorname{Rng}(T)],\) and verify Theorem 6.3.8. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) defined by \(T(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{rrr} 1 & -2 & 1 \\ 2 & -3 & -1 \\ 5 & -8 & -1 \end{array}\right].$$

Describe the transformation of \(\mathbb{R}^{2}\) with the given matrix as a product of reflections, stretches, and shears. $$A=\left[\begin{array}{ll}1 & 2 \\\0 & 1\end{array}\right]$$

Show that the given mapping is a nonlinear transformation. $$\begin{aligned} &T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R} \text { defined by }\\\ &T\left(a+b x+c x^{2}\right)=a+b+c+1 \end{aligned}$$.

Decide whether or not the given mapping \(T\) is a linear transformation. Justify your answers. For each mapping that is a linear transformation, decide whether or not \(T\) is one-to-one, onto, both, or neither, and find a basis and dimension for \(\operatorname{Ker}(T)\) and \(\operatorname{Rng}(T)\) $$\begin{array}{l} T: \mathbb{R}^{3} \rightarrow P_{2}(\mathbb{R}) \text { defined by } \\ \qquad T((a, b, c))=a x^{2}+(2 b-c) x+(a-2 b+c) \end{array}$$

Consider the linear transformation \(S: M_{n}(\mathbb{R}) \rightarrow\) \(M_{n}(\mathbb{R})\) defined by \(S(A)=A+A^{T},\) where \(A\) is a fixed \(n \times n\) matrix. (a) Find \(Ker(S)\) and describe it. What is dim \([Ker(S)]\)? (b) In the particular case when \(A\) is a \(2 \times 2\) matrix, determine a basis for \(\operatorname{Ker}(S),\) and hence, find its dimension.

Determine \(T(\mathbf{v})\) for the given linear transformation \(T\) and vector in \(V\) by (a) Computing \([T]_{B}^{C}\) and \([\mathbf{v}]_{B}\) and using Theorem 6.5 .4 (b) Direct calculation. \(T: P_{3}(\mathbb{R}) \rightarrow \mathbb{R}\) via \(T(p(x))=p(2),\) relative to the standard bases \(B\) and \(C ; p(x)=2 x-3 x^{2}\).

An invertible linear transformation \(\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is given. Find a formula for the inverse linear transformation. \(T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T_{1}(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{rr} -4 & -1 \\ 2 & 2 \end{array}\right]$$

If \(T: V \rightarrow W\) is an invertible linear transformation (that is, \(T^{-1}\) exists), show that \(T^{-1}: W \rightarrow V\) is also a linear transformation.

Prove that if \(T: V \rightarrow V\) is a one-to-one linear transformation, and \(V\) is finite-dimensional, then \(T^{-1}\) exists.