?In Exercises 7–8, find the domain and codomain of the transformation \(T\) defined by

In Exercises 1–2, find the domain and codomain of the transformation \(T_{A}(\mathbf{x})=A \mathbf{x}\). a. \(A\) has size \(3 \times 2\). b. \(A\) has size \(2 \times 3\). c. \(A\) has size \(3 \times 3\). d. \(A\) has size \(1 \times 6\). Equation Transcription: (????)=???? Text Transcription: T_A(????)=A???? A 3 x 2 A 2 x 3 A 3 x 3 A 1 x 6

In Exercises 1–2, find the domain and codomain of the transformation \(T_{A}(\mathbf{x})=A \mathbf{x}\). a. \(A\) has size \(4 \times 5\). b. \(A\) has size \(5 \times 4\). c. \(A\) has size \(4 \times 4\). d. \(A\) has size \(3 \times 1\). Equation Transcription: (????)=???? Text Transcription: T_A(????)=A???? A 4 x 5 A 5 x 4 A 4 x 4 A 3 x 1

In Exercises 3–4, find the domain and codomain of the transformation defined by the equations. a. \(\begin{aligned}&w_{1}=4 x_{1}+5 x_{2} \\&w_{2}=x_{1}-8 x_{2}\end{aligned}\) b. \(\begin{aligned}&w_{1}=5 x_{1}-7 x_{2} \\&w_{2}=6 x_{1}+x_{2} \\&w_{3}=2 x_{1}+3 x_{2}\end{aligned}\) Equation Transcription: Text Transcription: w_1=4x_1+5x_2 w_2=x_1-8x_2 w_1=5x_1+7x_2 w_2=6x_1+x_2 w_3=2x_1+3x_2

In Exercises 3–4, find the domain and codomain of the transformation defined by the equations. a. \(\begin{aligned}w_{1} &=x_{1}-4 x_{2}+8 x_{3} \\w_{2} &=-x_{1}+4 x_{2}+2 x_{3} \\w_{3} &=-3 x_{1}+2 x_{2}-5 x_{3}\end{aligned}\) b. \(\begin{aligned}&w_{1}=2 x_{1}+7 x_{2}-4 x_{3} \\&w_{2}=4 x_{1}-3 x_{2}+2 x_{3}\end{aligned}\) Equation Transcription: Text Transcription: w_1=x_1-4x_2+8x_3 w_2=-x_1+4x_2+2x_3 w_3=3x_1+2x_2-5x_3 w_1=2xv1+7x_2-4x_3 w_2=4x_1-3x_2+2x_3

In Exercises 5–6, find the domain and codomain of the transformation defined by the matrix product. a. \(\left[\begin{array}{lll}3 & 1 & 2 \\6 & 7 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2} \\x_{3}\end{array}\right]\) b. \(\left[\begin{array}{rr}2 & -1 \\4 & 3 \\2 & -5\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [_6 7 1 ^3 1 2] [ _x_3 ^x_2 ^x_1] [_2 -5 4 ^3 2 -1] [_x_2 ^x_1]

In Exercises 5–6, find the domain and codomain of the transformation defined by the matrix product. a. \(\left[\begin{array}{rr}6 & 3 \\-1 & 7\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]\) b. \(\left[\begin{array}{rrr}2 & 1 & -6 \\3 & 7 & -4 \\1 & 0 & 3\end{array}\right]\left[\begin{array}{l}x_{1}\\x_{2} \\x_{3}\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [_-1 7 ^6 3] [_x_2 ^x_1] [_1 0 3 ^3 7-4 ^2 1 -6] [ _x_3 ^x_2 ^x_1]

In Exercises 7–8, find the domain and codomain of the transformation \(T\) defined by the formula. a. \(T\left(x_{1}, x_{2}\right)=\left(2 x_{1}-x_{2}, x_{1}+x_{2}\right)\) b. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(4 x_{1}+x_{2}, x_{1}+x_{2}\right)\) Equation Transcription: Text Transcription: T T(x_1,x_2)=(2x_1-x2,x_1+x_2) T(x_1,x_2,x_3)=(4x_1-x_2,x_1+x_2)

In Exercises 7–8, find the domain and codomain of the transformation \(T\) defined by the formula. a. \(T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(x_{1}, x_{2}\right)\) b. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{2}-x_{3}, x_{2}\right)\) Equation Transcription: Text Transcription: T T(x_1,x_2,x_3,x_4)=(x_1,x_2) T(x_1,x_2,x_3)=(x_1,x_2-x_3,x_2)

In Exercises 9–10, find the domain and codomain of the transformation \(T\) defined by the formula. \(T\left(\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]\right)=\left[\begin{array}{c}4 x_{1} \\x_{1}-x_{2} \\3 x_{2}\end{array}\right]\) Equation Transcription: ([])= [] Text Transcription: T T([_x_2 ^x_1])= [_3x_2 ^x_1-x_2 ^4x_1]

In Exercises 9–10, find the domain and codomain of the transformation \(T\) defined by the formula. \(T\left(\left[\begin{array}{l}x_{1} \\x_{2} \\x_{3}\end{array}\right]\right)=\left[\begin{array}{c}x_{1} \\x_{2} \\x_{1}-x_{3} \\0\end{array}\right]\) Equation Transcription: ([])=[] Text Transcription: T T([ _x_3 ^x_2 ^x_1])=[_0 ^x_1-x_3 ^x_2 ^x_1]

In Exercises 11–12, find the standard matrix for the transformation defined by the equations. a. \(\begin{aligned}&w_{1}=2 x_{1}-3 x_{2}+x_{3} \\&w_{2}=3 x_{1}+5 x_{2}-x_{3}\end{aligned}\) b. \(\begin{aligned}&w_{1}=7 x_{1}+2 x_{2}-8 x_{3} \\&w_{2}=-x_{2}+5 x_{3} \\&w_{3}=-4 x_{1}+7x_{2}-x_{3}\end{aligned}\) Equation Transcription: Text Transcription: w_1=2x_1-3x_2+x_3 w_2=3x_1+5x_2-x_3 w_1=7x_1+2x2_-8x_3 w_2=-x_2+5x_3 w_3=-4x_1+7x_2-x_3

In Exercises 11–12, find the standard matrix for the transformation defined by the equations. a. \(\begin{aligned}&w_{1}=-x_{1}+x_{2} \\&w_{2}=3 x_{1}-2 x_{2} \\&w_{3}=5 x_{1}-7 x_{2}\end{aligned}\) b. \(\begin{aligned}&w_{1}=-x_{1} \\&w_{2}=x_{1}+x_{2} \\&w_{3}=x_{1}+x_{2}+x_{3} \\&w_{4}=x_{1}+x_{2}+x_{3}+x_{4}\end{aligned}\) Equation Transcription: Text Transcription: w_1=-x_1+x_2 w_2=3x_1-2x_2 w_3=5x_1-7x_2 w_1=-x_1 w_2=x1+x_2 w_3=x1+x_2+x_3 w_4=x1+x_2+x_3+x_4

Find the standard matrix for the transformation \(T\) defined by the formula. a. \(T\left(x_{1}, x_{2}\right)=\left(x_{2},-x_{1}, x_{1}+3 x_{2}, x_{1}-x_{2}\right)\) b. \(T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(7 x_{1},+2 x_{2}-x_{3}+x_{4}, x_{2}+x_{3},-x_{1}\right)\) c. \(T\left(x_{1}, x_{2}, x_{3}\right)=(0,0,0,0,0)\) d. \(T\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=\left(x_{4}, x_{1}, x_{3}, x_{2}, x_{1}-x_{3}\right)\) Equation Transcription: Text Transcription: T T(x_1,x_2)=(x_2,-x_1,x_1+3x_2,x_1-x_2) T(x_1,x_2,x_3,x_4)=(7x_1,+2x_2-x_3+x_4,x_2+x_3,-x_1) T(x_1,x_2,x_3)=(0,0,0,0,0) T(x_1,x_2,x_3,x_4)=(x_4,x_1,x_3,x_2,x_1-x_3)

Find the standard matrix for the operator \(T\) defined by the formula. a. \(T\left(x_{1}, x_{2}\right)=\left(2 x_{1}-x_{2}, x_{1}+x_{2}\right)\) b. \(T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{2}\right)\) c. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+2 x_{2}+x_{3}, x_{1}+5 x_{2}, x_{3}\right)\) d. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(4 x_{1}, 7 x_{2},-8 x_{3}\right)\) Equation Transcription: Text Transcription: T T(x_1,x_2)=(2x_1,-x_2,x_1+x_2) T(x_1,x_2)=(x_1,x_2) T(x_1,x_2)=(x_1,+2x_2+x_3,x_1+5x_2,x_3) T(x_1,x_2,x_3)=(4x_1,7x_2,-8x_3)

Find the standard matrix for the operator \(T: R^{3} \rightarrow R^{3}\) defined by \(\begin{aligned}&w_{1}=3 x_{1}+5 x_{2}-x_{3} \\&w_{2}=4 x_{1}-x_{2}+x_{3} \\&w_{3}=3 x_{1}+2x_{2}-x_{3}\end{aligned}\) and then compute \(T(-1,2,4)\) by directly substituting in the equations and then by matrix multiplication. Equation Transcription: Text Transcription: T:R^3 rightarrow R^3 w_1=3x_1+5x_2-x_3 w_2=4x_1-x_2+x_3 w_3=3x_1+2x_2-x_3 T(-1,2,4)

Find the standard matrix for the transformation \(T: R^{4} \rightarrow R^{2}\) defined by \(\begin{aligned}&w_{1}=2 x_{1}+3 x_{2}-5 x_{3}-x_{4} \\&w_{2}=x_{1}-5 x_{2}+2 x_{3}-2 x_{4}\end{aligned}\) and then compute \(T(1,-1,2,4)\) by directly substituting in the equations and then by matrix multiplication. Equation Transcription: Text Transcription: T:R^4 rightarrow R^2 w_1=2x_1+3x_2-5x_3-x_4 w_2=x_1-5x_2+2x_3-2x_4 T(1,-1,2,4)

In Exercises 17-18, find the standard matrix for the transformation and use it to compute \(T(\mathbf{x})\). Check your result by substituting directly in the formula for \(T\). a. \(T\left(x_{1}, x_{2}\right)=\left(-x_{1}+x_{2}, x_{2}\right) ; \mathbf{x}=(-1,4))\) b.\(T\left(x_{1}, x_{2}, x_{3}\right)=\left(2 x_{1}-x_{2}+x_{3}, x_{2}+x_{3}, 0\right) ; \mathrm{x}=(2,1,-3)\) Equation Transcription: (????) ???? ???? = Text Transcription: T(x) T T(x_1,x_2)=(-x_1+x_2,x_2);x=(-1,4) T(x_1,x2,x_3)=(2x_1-x_2+x_3,x_2+x_3,0);x=(2,1,-3)

In Exercises 17-18, find the standard matrix for the transformation and use it to compute \(T(\mathbf{x})\). Check your result by substituting directly in the formula for \(T\). a. \(T\left(x_{1}, x_{2}\right)=\left(2 x_{1}-x_{2}, x_{1}+x_{2}\right) ; \mathbf{x}=(-2,2)\) b. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{2}-x_{3}, x_{2}\right) ; \mathbf{x}=(1,0,5)\) Equation Transcription: (????) ???? ???? = Text Transcription: T(x) T T(x_1,x_2)=(2x_1+x_2,x_2+x_2);x=(-2,2) T(x_1,x2,x_3)=(x_1-x_2-x_3,x_2);x=(1,0,5) ???? ???? =

In Exercises 19-20, find \(T_{A}(\mathbf{x})\), and express your answer in matrix form. a. \(A=\left[\begin{array}{ll}1 & 2 \\3 & 4\end{array}\right] ; \mathbf{x}=\left[\begin{array}{r}3 \\-2\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}-1 & 2 & 0 \\3 & 1 & 5\end{array}\right] ; \mathbf{x}=\left[\begin{array}{r}-1 \\1 \\3\end{array}\right]\) Equation Transcription: (????) [];????=[] [];????=[] Text Transcription: TA(x) A=[_3 4 ^1 2];x=[_-2 ^1] A=[ _3 1 5 ^-1 2 0];x=[ _3 ^1 ^-1]

In Exercises 19-20, find \(T_{A}(\mathbf{x})\), and express your answer in matrix form. a. \(A=\left[\begin{array}{rrr}-2 & 1 & 4 \\3 & 5 & 7 \\6 & 0 & -1\end{array}\right] ;\mathbf{x}=\left[\begin{array}{l}x_{1} \\x_{2} \\x_{3}\end{array}\right]\) b. \(A=\left[\begin{array}{rr}-1 & 1 \\2 & 4 \\7 & 8\end{array}\right] ; \mathbf{x}=\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]\) Equation Transcription: (????) [];????=[] [];????=[] Text Transcription: TA(x) A=[ _6 0 -1 ^3 5 7 ^-2 1 4];x=[_ x_3 ^x_2 ^x_1] A=[ 7 8 2 4-1 1];x=[_x_2 ^x_1]

In Exercises 21-22, use Theorem 1.8.2 to show that \(T\) is a matrix transformation. a. \(T(x, y)=(2 x+y, x-y)\) b. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{3}, x_{1}+x_{2}\right)\) Equation Transcription: Text Transcription: T T(x,y)=(2x+y,x-y) T(x_1,x_2,x_3)=(x_1,x_3,x_1+x_2)

In Exercises 21-22, use Theorem 1.8.2 to show that \(T\) is a matrix transformation. a. \(T(x, y, z)=(x+y, y+z, x)\) b. \(T\left(x_{1}, x_{2}\right)=\left(x_{2}, x_{1}\right)\) Equation Transcription: Text Transcription: T T(x,y,z)=(x+y,y+z,x) T(x_1,x_2)=(x_2,x_1)

In Exercises 23-24, use Theorem 1.8.2 to show that \(T\) is not a matrix transformation. a. \(T(x, y)=\left(x^{2}, y\right)\) b. \(T(x, y, z)=(x, y, x z)\) Equation Transcription: Text Transcription: T T(x,y)=(x^2,y) T(x,y)=(x,y,xz)

In Exercises 23-24, use Theorem 1.8.2 to show that \(T\) is not a matrix transformation. a. \(T(x, y)=(x, y+1)\) b. \(T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{2}, \sqrt{x_{3}}\right)\) Equation Transcription: Text Transcription: T T(x,y)=(x,y+1) T(x_1,x_2,x_3)=(x_1,x_2,sqrt x_3)

A function of the form \(f(x)=m x+b\) is commonly called a "linear function" because the graph of \(y=m x+b\) is a line. Is \(f\) a matrix transformation on \(R\)? Equation Transcription: Text Transcription: f(x)=mx+b y=mx+b f R

Show that \(T(x, y)=(0,0)\) defines a matrix operator on \(R^{2}\) but \(T(x, y)=(1,1)\) does not. Equation Transcription: Text Transcription: T(x,y)=(0,0) R^2 T(x,y)=(1,1)

In Exercises 27-28, the images of the standard basis vectors for \(R^{3}\) are given for a linear transformation \(T: R^{3} \rightarrow R^{3\)}. Find the standard matrix for the transformation, and find \(T(x)\). \(T\left(\mathbf{e}_{1}\right)=\left[\begin{array}{l}1 \\3 \\0\end{array}\right], T\left(\mathbf{e}_{2}\right)=\left[\begin{array}{l}0 \\0 \\1\end{array}\right],T\left(\mathbf{e}_{3}\right)=\left[\begin{array}{r}4 \\-3 \\-1\end{array}\right] ; \quad \mathbf{x}=\left[\begin{array}{l}2 \\1 \\0\end{array}\right]\) Equation Transcription: (????1)=[],(????2)=[],(????3)=[]; ????=[] Text Transcription: R^3 T:R^3 rightarrow R^3 T(x) T(e_1)=[_ 0 ^3 ^1],T(e_2)=[_1 ^0 ^0],T(e_3)=[ _-1 ^-3 ^4]; x=[_ 0 ^1 ^2]

In Exercises 27-28, the images of the standard basis vectors for \(R^{3}\) are given for a linear transformation \(T: R^{3} \rightarrow R^{3\)}. Find the standard matrix for the transformation, and find \(T(x)\). \(T\left(\mathbf{e}_{1}\right)=\left[\begin{array}{l}2 \\1 \\3\end{array}\right],T\left(\mathbf{e}_{2}\right)=\left[\begin{array}{r}-3 \\-1 \\0\end{array}\right],T\left(\mathbf{e}_{3}\right)=\left[\begin{array}{l}1 \\0 \\2\end{array}\right] ;\quad \mathbf{x}=\left[\begin{array}{l}3 \\2 \\1\end{array}\right]\) Equation Transcription: (????1)=[],(????2)=[],(????3)=[]; ????=[] Text Transcription: R^3 T:R^3 rightarrow R^3 T(x) T(e_1)=[_ 3 ^1 ^2],T(e_2)=[_0 ^-1 ^*3],T(e_3)=[ _2 ^0 ^1]; x=[_ 1 ^2 ^3]

Use matrix multiplication to find the reflection of (-1,2) about the a. \(x-\text { axis }\). b. \(y-\text { axis }\). c. \(line y=x\). Equation Transcription: Text Transcription: x-axis y-axis y=x

Use matrix multiplication to find the reflection of \((a,b)\) about the a. \(x-\text { axis }\). b. \(y-\text { axis }\). c. \(line y=x\). Equation Transcription: Text Transcription: (a,b) x-axis y-axis y=x

Use matrix multiplication to find the reflection of (2,-5,3)about the a. \(x y \text { - plane }\) b. \(x z \text { - plane }\) c. \(y z \text {-plane }\) Equation Transcription: Text Transcription: xy-plane xz-plane yz-plane

Use matrix multiplication to find the reflection of \((a,b,c)\) about the a. \(x y \text { - plane }\) b. \(x z \text { - plane }\) c. \(y z \text {-plane }\) Equation Transcription: Text Transcription: (a,b,c) xy-plane xz-plane yz-plane

Use matrix multiplication to find the orthogonal projection of (2,-5) onto the a. \(x-\text { axis }\). b. \(y-\text { axis }\). Equation Transcription: Text Transcription: x-axis y-axis

Use matrix multiplication to find the orthogonal projection of \((a, b)\) onto the a. \(x-\text { axis }\). b. \(x\y-\text { axis }\). Equation Transcription: Text Transcription: (a,b) x-axis y-axis

Use matrix multiplication to find the orthogonal projection of (-2,1,3) onto the a. \(x y \text {-plane }\). b. \(x z \text {-plane }\). c. \(y z \text {-plane }\). Equation Transcription: Text Transcription: xy-plane xz-plane yz-plane

Use matrix multiplication to find the orthogonal projection of \((a,b,c)\) onto the a. \(x y \text {-plane }\). b. \(x z \text {-plane }\). c. \(y z \text {-plane }\). Equation Transcription: Text Transcription: (a,b,c) xy-plane xz-plane yz-plane

Use matrix multiplication to find the image of the vector (3,-4) when it is rotated about the origin through an angle of a. \(\theta=30^{\circ}\) b.\(\theta=-60^{\circ}\) c. \(\theta=45^{\circ}\) d. \(\theta=90^{\circ}\) Equation Transcription: Text Transcription: theta=30^degree theta=-60^degree theta=45^degree theta=90^degree

Use matrix multiplication to find the image of the nonzero vector \(v=\left(v_{1}, v_{2}\right)\) when it is rotated about the origin through a. a positive angle \(\alpha\). b. a negative angle \(-\alpha\). Equation Transcription: Text Transcription: v=(v_1,v_2) alpha -alpha

Let \(T: R^{2} \rightarrow R^{2}\) be a linear operator for which the images of the standard basis vectors for \(R^{2}\) are \(T\left(\mathbf{e}_{1}\right)=(a, b)\) and \(T\left(\mathbf{e}_{2}\right)=(c, d)\). Find \(T(1,1)\). Equation Transcription: (????1) (????2) Text Transcription: T:R^2 rightarrow R^2 R^2 T(e_1)=(a,b) T(e_2)=(c,d) T(1,1)

Let \(T_{A}: R^{2} \rightarrow R^{2}\) be multiplication by \(A=\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\) and let \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) be the standard basis vectors for \(R^{2}\). Find the following vectors by inspection. a. \(T_{A}\left(k \mathbf{e}_{1}\right)\) b. \(T_{A}\left(k \mathbf{e}_{1}+l \mathbf{e}_{2}\right)\) Equation Transcription: [] ????1 ????2 ????1 ????1????2 Text Transcription: T_A:R^2 rightarrow R^2 A=[_c d ^a b] e_1 e_2 R^2 T_A(ke_1) T_A(ke_1+le_2)

Let \(T_{A}: R^{3} \rightarrow \(R^{3}\) be multiplication by \(A=\left[\begin{array}{rcc}-1 & 3 & 0 \\2 & 1 & 2 \\4 & 5 & -3\end{array}\right]\) and let \(\mathbf{e}_{1}\), \(\mathrm{e}_{2}\) and \(\mathrm{e}_{3}\) be the standard basis vectors for \(R^{3}\). Find the following vectors by inspection. a. \(T_{A}\left(\mathbf{e}_{1}\right), T_{A}\left(\mathbf{e}_{2}\right)\) and \(T_{A}\left(\mathbf{e}_{3}\right)\) b. \(T_{A}\left(\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}\right)\) c. \(T_{A}\left(7 \mathbf{e}_{3}\right)\) Equation Transcription: [] ????1 ????2 ????3 (????1),(????2) (????3) (????1+????2+????3) (????3) Text Transcription: TA:R^3 rightarrow R^3 A=[ _4 5 -3 ^2 1 2 ^-1 3 0] e_1 e_2 e_3 R^3 T_A(e_1),T_A(e_2) T_A(e_3) T_A(e_1+e_2+e_3) T_A(7e_3)

For each orthogonal projection operator in Table 4 use the standard matrix to compute \(T(1,2,3)\), and convince yourself that your result makes sense geometrically. Equation Transcription: Text Transcription: T(1,2,3)

For each reflection operator in Table 2 use the standard matrix to compute \(T(1,2,3)\), and convince yourself that your result makes sense geometrically. Equation Transcription: Text Transcription: T(1,2,3)

If multiplication by \(A\) rotates a vector \(x\) in the \(x y \text {-plane }\) through an angle \(\theta\), what is the effect of multiplying \(x by \(A^{T}\)? Explain your reasoning. Equation Transcription: Text Transcription: A x xy-plane x A^T

Find the standard matrix \(A\) for the linear transformation \(T: R^{2} \rightarrow R^{2}\) for which \(T\left(\left[\begin{array}{l}1 \\1\end{array}\right]\right)=\left[\begin{array}{r}1 \\-2\end{array}\right], T\left(\left[\begin{array}{l}2 \\3\end{array}\right]\right)=\left[\begin{array}{r}-2 \\5\end{array}\right]\) Equation Transcription: ([])=[],([])=[] Text Transcription: A T:R^2 rightarrow R^2 T([_1 ^1])=[_-2 ^1],T([_3 ^2])=[ _5 ^-2]

Find the standard matrix \(A\) for the linear transformation \(T: R^{3} \rightarrow R^{3}\) for which \(T\left(\left[\begin{array}{l}1 \\0 \\2\end{array}\right]\right)=\left[\begin{array}{r}2 \\-3 \\10\end{array}\right], T\left(\left[\begin{array}{l}1 \\1 \\1\end{array}\right]\right)=\left[\begin{array}{l}1 \\3 \\8\end{array}\right], T\left(\left[\begin{array}{r} -3 \\-1 \\2\end{array}\right]\right)=\left[\begin{array}{r}-5 \\-11 \\7\end{array}\right]\) Equation Transcription: ([])=[],([])=[],([])=[] Text Transcription: A T:R^2 rightarrow R^2 T([ _2 ^0 ^1])=[_10 ^-3 ^2],T([_1 ^1 ^1])=[ _8 ^3 ^1],T([ _2 ^-1 ^-3])=[ _7 ^-1 ^-5]

Let \(x_{0}\) be a nonzero column vector in \(R^{2}\), and suppose that \(T: R^{2} \rightarrow \(R^{2}\) is the transformation defined by the formula \(T(x)=x_{0}+R_{\theta} x\), where \(R_{\theta}\) is the standard matrix of the rotation of \(R^{2}\) about the origin through the angle \(\theta\). Give a geometric description of this transformation. Is it a matrix transformation? Explain. Equation Transcription: Text Transcription: x_0 R2 T:RR2 rightarrow R^2 T(x)=x_0+R_theta x R_theta R^2 theta

In each part of the accompanying figure, find the standard matrix for the pictured operator.

In a sentence, describe the geometric effect of multiplying a vector \(\mathrm{x}\) by the matrix \(A=\left[\begin{array}{cc}\cos ^{2} \theta-\sin ^{2} \theta & -2 \sin \theta \cos \theta \\2 \sin \theta\cos \theta & \cos ^{2} \theta-\sin ^{2} \theta\end{array}\right]\) Equation Transcription: ???? [] Text Transcription: x A=[_2 sin theta cos theta cos^2 theta -sin^2 theta ^cos^2 theta-sin^2 theta -2 sin theta cos theta ]

a. Prove: If \(T: R^{n} \rightarrow R^{m}\) is a matrix transformation, then \(T(0)=0\); that is, \(T\) maps the zero vector in \(R^{n}\) into the zero vector in \(R^{m}\). Equation Transcription: Text Transcription: T:R^nR^m T(0)=0 T R^n R^m

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. a. If \(A\) is a \(2 \times 3\) matrix, then the domain of the transformation \(T_{A}\) is \(R^{2}\). Equation Transcription: Text Transcription: A 2 x 3 T_A R^2

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. b. If \(A\) is an \(m \times n\) matrix, then the codomain of the transformation \(T_{A}\) is \(R^{n}\). Equation Transcription: Text Transcription: A m x n T_A R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. There is at least one linear transformation \(T: R^{n} \rightarrow R^{m}\) for which \(T(2 x)=4 T(x)\) for some vector \(x\) in \(R^{n}\) Equation Transcription: Text Transcription: T:R^nR^m T(2x)=4T(x) x R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. d. There are linear transformations from \(R^{n}\) to \(R^{m}\) that are not matrix transformations. Equation Transcription: Text Transcription: R^n R^m

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. e. If \(T_{A}: R^{n} \rightarrow R^{n}\) and if \(T_{A}(x)=0\) for every vector \(x\) in \(R^{n}\), then \(A\) is the \(n \times n\) zero matrix. Equation Transcription: Text Transcription: TA:R^nR^n T_A(x)=0 x R^n A n x n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. f. There is only one matrix transformation \(T: R^{n} \rightarrow R^{m}\) such that \(T(-x)=-T(x)\) for every vector \(x\) in \(R^{n}\) Equation Transcription: Text Transcription: T:R^nR^m T(-x)=-T(x) x R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. g. If \(b\) is a nonzero vector in \(R^{n}\), then \(T(x)=x+b\) is a matrix operator on \(R^{n}\). Equation Transcription: Text Transcription: b R^n T(x)=x+b R^n

a. Prove: If \(T: R^{n} \rightarrow R^{m}\) is a matrix transformation, then \(T(0)=0\); that is, \(T\) maps the zero vector in \(R^{n}\) into the zero vector in \(R^{m}\). Equation Transcription: Text Transcription: T:R^nR^m T(0)=0 T R^n R^m

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. a. If \(A\) is a \(2 \times 3\) matrix, then the domain of the transformation \(T_{A}\) is \(R^{2}\). Equation Transcription: Text Transcription: A 2 x 3 T_A R^2

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. b. If \(A\) is an \(m \times n\) matrix, then the codomain of the transformation \(T_{A}\) is \(R^{n}\). Equation Transcription: Text Transcription: A m x n T_A R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. There is at least one linear transformation \(T: R^{n} \rightarrow R^{m}\) for which \(T(2 x)=4 T(x)\) for some vector \(x\) in \(R^{n}\) Equation Transcription: Text Transcription: T:R^nR^m T(2x)=4T(x) x R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. d. There are linear transformations from \(R^{n}\) to \(R^{m}\) that are not matrix transformations. Equation Transcription: Text Transcription: R^n R^m

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. e. If \(T_{A}: R^{n} \rightarrow R^{n}\) and if \(T_{A}(x)=0\) for every vector \(x\) in \(R^{n}\), then \(A\) is the \(n \times n\) zero matrix. Equation Transcription: Text Transcription: TA:R^nR^n T_A(x)=0 x R^n A n x n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. f. There is only one matrix transformation \(T: R^{n} \rightarrow R^{m}\) such that \(T(-x)=-T(x)\) for every vector \(x\) in \(R^{n}\) Equation Transcription: Text Transcription: T:R^nR^m T(-x)=-T(x) x R^n

In parts (a)–(g) determine whether the statement is true or false, and justify your answer. g. If \(b\) is a nonzero vector in \(R^{n}\), then \(T(x)=x+b\) is a matrix operator on \(R^{n}\). Equation Transcription: Text Transcription: b R^n T(x)=x+b R^n