Chapter 6

Let \(T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) and \(T_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be the linear transformations with matrices $$A=\left[\begin{array}{rr} 1 & -1 \\ 3 & 2 \end{array}\right], \quad B=[-11]$$ respectively. Find \(T_{2} T_{1} .\) Does \(T_{1} T_{2}\) exist? Explain.

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 \(\operatorname{Rng}(T)\) $$\begin{aligned} &T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4} \text { defined by }\\\ &T(x, y)=(x+y, 0, x-y, x y) \end{aligned}$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: M_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R})\) given by $$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=(a-d)+3 b x^{2}+(c-a) x^{3}$$ (a) \(B=\left\\{E_{11}, E_{12}, E_{21}, E_{22}\right\\} ; C=\left\\{1, x, x^{2}, x^{3}\right\\}\) (b) \(B=\left\\{E_{21}, E_{11}, E_{22}, E_{12}\right\\} ; C=\left\\{x, 1, x^{3}, x^{2}\right\\}\)

Consider \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}\) defined by \(T(\mathbf{x})=A \mathbf{x}\) where \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 4 \\ 4 & 8 \\ 8 & 16\end{array}\right] .\) For each \(\mathbf{x}\) below, find \(T(\mathbf{x})\) and thereby determine whether \(\mathbf{x}\) is in \(\operatorname{Ker}(T).\) (a) \(\mathbf{x}=(-10,5).\) (b) \(\mathbf{x}=(1,-1).\) (c) \(\mathbf{x}=(2,-1).\)

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). $$A=\left[\begin{array}{rr}1 & -1 \\\1 & 2\end{array}\right]$$

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) defined by $$ T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+3 x_{2}+x_{3}, x_{1}-x_{2}\right) $$.

Let \(T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) and \(T_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear transformations with matrices $$A=\left[\begin{array}{rr} -1 & 2 \\ 3 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 5 \\ -2 & 0 \end{array}\right]$$ respectively. Find \(T_{1} T_{2}\) and \(T_{2} T_{1} .\) Does \(T_{1} T_{2}=T_{2} T_{1} ?\)

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 \(\operatorname{Rng}(T)\) $$\begin{aligned} &T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} \text { defined by }\\\ &T(x, y, z)=(2 x-3 y,-x) \end{aligned}$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}\) given by $$T\left(a+b x+c x^{2}\right)=(a-3 c, 2 a+b-2 c)$$ (a) \(B=\left\\{1, x, x^{2}\right\\} ; C=\\{(1,0),(0,1)\\}\) (b) \(B=\left\\{1,1+x, 1+x+x^{2}\right\\} ; C=\\{(1,-1),(2,1)\\}\)

Consider \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) defined by \(T(\mathbf{x})=A \mathbf{x},\) where \(A=\left[\begin{array}{lll}1 & -1 & 2 \\\ 1 & -2 & -3\end{array}\right] .\) For each \(\mathbf{x}\) below, find \(T(\mathbf{x})\) and thereby determine whether \(\mathbf{x}\) is in \(\operatorname{Ker}(T).\) (a) \(\mathbf{x}=(7,5,-1).\) (b) \(\mathbf{x}=(-21,-15,2).\) (c) \(\mathbf{x}=(35,25,-5).\)

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). $$A=\left[\begin{array}{rr}0 & 1 \\\\-1 & 0\end{array}\right]$$

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by $$ T\left(x_{1}, x_{2}\right)=\left(x_{1}+2 x_{2}, 2 x_{1}-x_{2}\right) $$.

Let \(T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) and \(T_{2}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) be the linear transformations with matrices $$A=\left[\begin{array}{rr} 2 & -1 \\ 0 & 1 \\ 1 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0 & -4 & 3 \\ 1 & 0 & 1 \end{array}\right]$$ respectively. Find \(T_{1} T_{2}, \operatorname{Ker}\left(T_{1} T_{2}\right), \operatorname{Rng}\left(T_{1} T_{2}\right), T_{2} T_{1}\) \(\operatorname{Ker}\left(T_{2} T_{1}\right),\) and \(\operatorname{Rng}\left(T_{2} T_{1}\right)\)

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 \(\operatorname{Rng}(T)\) $$\begin{aligned} &T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} \text { defined by }\\\ &T((x, y, z))=(-3 z, 2 x-y+5 z) \end{aligned}$$

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. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by \(T(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{ll} 3 & 6 \\ 1 & 2 \end{array}\right].$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: P_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R})\) given by $$T(p(x))=(x+1) p(x)$$ (a) \(B=\left\\{1, x, x^{2}\right\\} ; C=\left\\{1, x, x^{2}, x^{3}\right\\}\) (b) \(B=\left\\{1, x-1,(x-1)^{2}\right\\}\) \(C=\left(1, x-1,(x-1)^{2},(x-1)^{3}\right)\)

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). $$A=\left[\begin{array}{rr}-2 & -2 \\\\-2 & 0\end{array}\right]$$

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: C^{2}(I) \rightarrow C^{0}(I)\) defined by $$ T(y)=y^{\prime \prime}+a_{1} y^{\prime}+a_{2} y $$ where \(a_{1}\) and \(a_{2}\) are functions defined on \(I\).

Let \(T_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) and \(T_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear transformations with matrices $$A=\left[\begin{array}{ll} 1 & -1 \\ 2 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 3 & -1 \end{array}\right]$$ respectively. Find \(\operatorname{Ker}\left(T_{1}\right), \operatorname{Ker}\left(T_{2}\right), \operatorname{Ker}\left(T_{1} T_{2}\right),\) and \(\operatorname{Ker}\left(T_{2} T_{1}\right)\)

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 \(\operatorname{Rng}(T)\) $$T: C[0,1] \rightarrow \mathbb{R}^{2} \text { defined by } T(g)=(g(0), g(1))$$

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 $$A=\left[\begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & 2 \\ 2 & -1 & 1 \end{array}\right].$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: \mathbb{R}^{3} \rightarrow \operatorname{span}\\{\cos x, \sin x\\}\) given by $$T(a, b, c)=(a-2 c) \cos x+(3 b+c) \sin x$$ (a) \(B=\\{(1,0,0),(0,1,0),(0,0,1)\\}\) \(C=\\{\cos x, \sin x\\}\) (b) \(B=\\{(2,-1,-1),(1,3,5),(0,4,-1)\\}\) \(C=\\{\cos x-\sin x, \cos x+\sin x\\}\)

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). $$A=\left[\begin{array}{rl}1 & 1 \\\\-1 & 1\end{array}\right]$$

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: C^{2}(I) \rightarrow C^{0}(I)\) defined by $$ T(y)=y^{\prime \prime}-16 y $$.

Let \(T_{1}: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\) and \(T_{2}: M_{n}(\mathbb{R}) \rightarrow\) \(M_{n}(\mathbb{R})\) be the linear transformations defined by \(T_{1}(A)=A-A^{T}\) and \(T_{2}(A)=A+A^{T} .\) Show that \(T_{2} T_{1}\) is the zero transformation.

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 $$A=\left[\begin{array}{rrr} 1 & -2 & 1 \\ 2 & -3 & -1 \\ 5 & -8 & -1 \end{array}\right].$$

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 \(\operatorname{Rng}(T)\) $$T: \mathbb{R}^{2} \rightarrow \mathbb{R} \text { defined by } T(x, y)=\frac{x+y}{5}$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}\) given by $$T(A)=(\operatorname{tr}(A), \operatorname{tr}(A))$$ (a) \(B=\left\\{E_{11}, E_{12}, E_{21}, E_{22}\right\\} ; C=\\{(1,0),(0,1)\\}\) (b) \(B=\left\\{\left[\begin{array}{cc}-1 & -2 \\ -2 & -3\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 2 & 2\end{array}\right],\left[\begin{array}{cc}0 & -3 \\ 2 & -2\end{array}\right],\left[\begin{array}{ll}0 & 4 \\ 1 & 0\end{array}\right]\right\\}\) \(C=\\{(1,0),(0,1)\\}\)

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

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\) defined by $$ T(A)=A B-B A $$ where \(B\) is a fixed \(n \times n\) matrix.

Define \(T_{1}: C^{1}[a, b] \rightarrow C^{0}[a, b]\) and \(T_{2}\) \(C^{0}[a, b] \rightarrow C^{1}[a, b]\) by $$\begin{aligned} T_{1}(f) &=f^{\prime} \\ \left[T_{2}(f)\right](x) &=\int_{a}^{x} f(t) d t, \quad a \leq x \leq b \end{aligned}$$ (a) If \(f(x)=\sin (x-a),\) find \(\left[T_{1}(f)\right](x) \quad\) and \(\quad\left[T_{2}(f)\right](x)\) and show that, for the given function, $$\left[T_{1} T_{2}\right](f)=\left[T_{2} T_{1}\right](f)=f$$ (b) Show that for general functions \(f\) and \(g\) $$\left[T_{1} T_{2}\right](f)=f,\left\\{\left[T_{2} T_{1}\right](g)\right\\}(x)=g(x)-g(a)$$

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. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) defined by \(T(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ -3 & 3 & -6 \end{array}\right].$$

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 \(\operatorname{Rng}(T)\) $$T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2} \text { defined by } T\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=(a c, b d)$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: P_{3}(\mathbb{R}) \rightarrow P_{2}(\mathbb{R})\) given by \(T(p(x))=p^{\prime}(x)\) (a) \(B=\left\\{1, x, x^{2}, x^{3}\right\\} ; C=\left\\{1, x, x^{2}\right\\}\) (b) \(B=\left\\{x^{3}, x^{3}+1, x^{3}+x, x^{3}+x^{2}\right\\}\) \(C=\left\\{1,1+x, 1+x+x^{2}\right\\}\)

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

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: C^{0}[a, b] \rightarrow \mathbb{R}\) defined by $$ T(f)=\int_{a}^{b} f(x) d x $$.

Let \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) be a basis for the vector space \(V,\) and suppose that \(T_{1}: V \rightarrow V\) and \(T_{2}: V \rightarrow V\) are the linear transformations satisfying $$\begin{array}{ll} T_{1}\left(\mathbf{v}_{1}\right)=\mathbf{v}_{1}-\mathbf{v}_{2}, & T_{1}\left(\mathbf{v}_{2}\right)=2 \mathbf{v}_{1}+\mathbf{v}_{2} \\ T_{2}\left(\mathbf{v}_{1}\right)=\mathbf{v}_{1}+2 \mathbf{v}_{2}, & T_{2}\left(\mathbf{v}_{2}\right)=3 \mathbf{v}_{1}-\mathbf{v}_{2} \end{array}$$ Determine \(\left(T_{2} T_{1}\right)(\mathbf{v})\) for an arbitrary vector \(\mathbf{v}\) in \(V\)

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. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) defined by \(T(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{lll} 1 & 3 & 2 \\ 2 & 6 & 5 \end{array}\right].$$

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 \(\operatorname{Rng}(T)\) $$\begin{aligned} T: P_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R}) \text { defined by } \\ T\left(a+b x+c x^{2}\right)=\left[\begin{array}{cc} -a-b & 0 \\ 3 c-a & -2 b \end{array}\right] \end{aligned}$$

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

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: M_{n}(\mathbb{R}) \rightarrow \mathbb{R}\) defined by \(T(A)=\operatorname{tr}(A),\) where \(\operatorname{tr}(A)\) denotes the trace of \(A\).

Repeat Problem 7 under the assumption $$\begin{array}{ll} T_{1}\left(\mathbf{v}_{1}\right)=3 \mathbf{v}_{1}+\mathbf{v}_{2}, & T_{1}\left(\mathbf{v}_{2}\right)=\mathbf{0} \\ T_{2}\left(\mathbf{v}_{1}\right)=-5 \mathbf{v}_{2}, & T_{2}\left(\mathbf{v}_{2}\right)=-\mathbf{v}_{1}+6 \mathbf{v}_{2} \end{array}$$

Determine the matrix representation \([T]_{B}^{C}\) for the given linear transformation \(T\) and ordered bases \(\bar{B}\) and \(C\). \(T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})\) given by $$T(A)=2 A-A^{T}$$ (a) \(B=C=\left\\{E_{11}, E_{12}, E_{21}, E_{22}\right\\}\) (b) \(B=\left\\{\left[\begin{array}{ll}-1 & -2 \\ -2 & -3\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 2 & 2\end{array}\right],\left[\begin{array}{ll}0 & -3 \\ 2 & -2\end{array}\right],\left[\begin{array}{ll}0 & 4 \\ 1 & 0\end{array}\right]\right\\}\) \(C=\left\\{E_{11}, E_{12}, E_{21}, E_{22}\right\\}\)

Describe the transformation of \(\mathbb{R}^{2}\) with the given matrix as a product of reflections, stretches, and shears. $$A=\left[\begin{array}{ll}1 & 0 \\\3 & 1\end{array}\right]$$

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(S: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\) defined by $$ S(A)=A+A^{T} $$.

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 \(\operatorname{Rng}(T)\) $$\begin{array}{l} T: \mathbb{R}^{3} \rightarrow P_{2}(\mathbb{R}) \text { defined by } \\ \qquad T((a, b, c))=a x^{2}+(2 b-c) x+(a-2 b+c) \end{array}$$

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. \(T: \mathbb{R}^{3} \rightarrow P_{3}(\mathbb{R})\) via $$ T(a, b, c)=2 a-(a+b-c) x+(2 c-a) x^{3} $$ relative to the standard bases \(B\) and \(C ; \mathbf{v}=(2,-1,5)\)

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

Describe the transformation of \(\mathbb{R}^{2}\) with the given matrix as a product of reflections, stretches, and shears. $$A=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$

Find \(\operatorname{Ker}(T)\) and \(\operatorname{Rng}(T),\) and hence, determine whether the given transformation is one-to-one, onto, both, or neither. If \(T^{-1}\) exists, find it. $$T(\mathbf{x})=A \mathbf{x}, \text { where } A=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]$$