?a. Use Theorem 5.2.1 to show that the following matrix is

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. Transition matrices are invertible.

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. If \(B\) is a basis for a vector space \(R^{n}\), then \(P_{B \rightarrow B}\) is the identity matrix. Equation Transcription: Text Transcription: B R^n P_B rightarrow B

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. If \(P_{B_{1} \rightarrow B_{2}}\) is a diagonal matrix, then each vector in \(B_{2}\) is a scalar multiple of some vector in \(B_{1}\). Equation Transcription: Text Transcription: P_B_{1} rightarrow B_{2} B_2 B_1

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. If each vector in \(B_{2}\) is a scalar multiple of some vector in \(B_{1}\), then \(P_{B_{1} \rightarrow B_{2}}\) is a diagonal matrix. Equation Transcription: Text Transcription: B_2 B_1 P_B_{1} rightarrow B_{2}

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. If \(A\) is a square matrix, then \(A=P_{B_{1} \rightarrow B_{2}}\) for some bases \(B_{1}\) and \(B_{2}$ for $R^{n}\). Equation Transcription: Text Transcription: A A = P_B_{1} rightarrow B_{2} B_1 B_ R^n

Let $\(P=\left[\begin{array}{rrrr}5 & 8 & 6 & -13 \\3 & -1 & 0 & -9 \\0 & 1 & -1 & 0 \\2 & 4 & 3 & -5 \end{array}\right]\) and \(\begin{array}{ll}\mathbf{v}_{1}=(2,4,3,-5), & \mathbf{v}_{2}=(0,1,-1,0)\\\mathbf{v}_{3}=(3,-1,0,-9), & \mathbf{v}_{4}=(5,8,6,-13)\end{array}\) Find a basis \(B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}\right\}\) for \(R^{4}\) for which \(P\) is the transition matrix from \(B\) to \(B^{\prime}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}\). Equation Transcription: = [ ] Text Transcription: P = [5 & 8 & 6 & -13 \\3 & -1 & 0 & -9 \\0 & 1 & -1 & 0 \\2 & 4 & 3 & -5] v_1 = (2,4,3,-5), v_2 = (0,1,-1,0) v_3 = (3,-1,0,-9), v_4 = (5,8,6,-13) B = {u_1,u_2, u_3, u_4} R^4 P B B’ = {v_1, v_2, v_3, v_4}

Given that the matrix for a linear transformation \(T: R^{4} \rightarrow R^{4}\) relative to the standard basis \(B=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}, \mathbf{e}_{4}\right\}\) for \(R^{4}\) is \(\left[\begin{array}{rrrr}1 & 2 & 0 & 1 \\3 & 0 & -1 & 2 \\2 & 5 & 3 & 1 \\1 & 2 & 1 & 3\end{array}\right]\) find the matrix for \(T\) relative to the basis \(B^{\prime}=\left\{\mathbf{e}_{1},\mathbf{e}_{1}+\mathbf{e}_{2},\mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}, \mathbf{e}_{1}+\mathbf{e}_{2}+\mathbf{e}_{3}+\mathbf{e}_{4}\right\}\) Equation Transcription: [ ] Text Transcription: T: R^4 rightarrow R^4 B = {e_1, e_2, e_3, e_4} R^4 [1 & 2 & 0 & 1 \\3 & 0 & -1 & 2 \\2 & 5 & 3 & 1 \\1 & 2 & 1 & 3] T B’ = {e_1, e_1 + e_2, e_1 + e_2 + e_3, e_1 + e_2 + e_3 + e_4}

In Exercises 1–2, express the product Ax as a linear combination of the column vectors of \(A\). a. \(\left[\begin{array}{rr}2 & 3 \\ -1 & 4\end{array}\right]\left[\begin{array}{l}1 \\ 2\end{array}\right]\) b. \(\left[\begin{array}{rrr}4 & 0 & -1 \\ 3 & 6 & 2 \\ 0 & -1 & 4\end{array}\right]\left[\begin{array}{r}-2 \\ 3 \\ 5\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A [2 & 3 \\ -1 & 4] [1 \\ 2] [4 & 0 & -1 \\ 3 & 6 & 2 \\ 0 & -1 & 4] [-2 \\ 3 \\ 5]

In Exercises 1–2, express the product Ax as a linear combination of the column vectors of \(A\). a. \(\left[\begin{array}{rrr}-3 & 6 & 2 \\ 5 & -4 & 0 \\ 2 & 3 & -1 \\ 1 & 8 & 3\end{array}\right]\left[\begin{array}{r}-1 \\ 2 \\ 5\end{array}\right]\) b. \(\left[\begin{array}{rrr}2 & 1 & 5 \\ 6 & 3 & -8\end{array}\right]\left[\begin{array}{r}3 \\ 0 \\ -5\end{array}\right]\) Equation Transcription: [] [] [] Text Transcription: A [-3 & 6 & 2 \\ 5 & -4 & 0 \\ 2 & 3 & -1 \\ 1 & 8 & 3\] [-1 \\ 2 \\ 5] [2 & 1 & 5 \\ 6 & 3 & -8] [3 \\ 0 \\ -5]

In Exercises 3-4, determine whether \(\mathbf{b}\) is in the column space of \(A\), and if so, express \(\mathbf{b}\) as a linear combination of the column vectors of \(A\). a. \(A=\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 3\end{array}\right] ; \quad \mathbf{b}=\left[\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & -1 & 1 \\ 9 & 3 & 1 \\ 1 & 1 & 1\end{array}\right] ; \quad \mathbf{b}=\left[\begin{array}{r}5 \\ 1 \\ -1\end{array}\right]\) Equation Transcription: []; [] []; [] Text Transcription: b A b A A = [1 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 3\] ; b = [-1 \\ 0 \\ 2] A = [1 & -1 & 1 \\ 9 & 3 & 1 \\ 1 & 1 & 1] ; b = [5 \\ 1 \\ -1]

In Exercises 3-4, determine whether \(\mathbf{b}\) is in the column space of \(A\), and if so, express \(\mathbf{b}\) as a linear combination of the column vectors of \(A\). a. \(=\left[\begin{array}{rrr}1 & -1 & 1 \\ -1 & 1 & -1 \\ -1 & -1 & 1\end{array}\right] ; \quad \mathbf{b}=\left[\begin{array}{l}2 \\ 0 \\ 0\end{array}\right]\) b. \(A=\left[\begin{array}{llll}1 & 2 & 0 & 1 \\ 0 & 1 & 2 & 1 \\ 1 & 2 & 1 & 3 \\ 0 & 1 & 2 & 2\end{array}\right] ; \quad \mathbf{b}=\left[\begin{array}{l}4 \\ 3 \\ 5 \\ 7\end{array}\right]\) Equation Transcription: []; [] = [ ];= [ ] Text Transcription: b A b A A = [1 & -1 & 1 \\ -1 & 1 & -1 \\ -1 & -1 & 1] ; b = [2 \\ 0 \\ 0] A = [1 & 2 & 0 & 1 \\ 0 & 1 & 2 & 1 \\ 1 & 2 & 1 & 3 \\ 0 & 1 & 2 & 2] ; b = [4 \\ 3 \\ 5 \\ 7]

Suppose that \(x_{1}=3, x_{2}=0, x_{3}=-1, x_{4}=5\) is a solution of a nonhomogeneous linear system \(A \mathbf{x}=\mathbf{b}\) and that the solution set of the homogeneous system \(A \mathbf{x}=\mathbf{0}\) is given by the formulas \(x_{1}=5 r-2 s, \quad x_{2}=s, \quad x_{3}=s+t, \quad x_{4}=t\) a. Find a vector form of the general solution of \(A \mathrm{x}=0\). b. Find a vector form of the general solution of \(A \mathbf{x}=\mathbf{b}\). Equation Transcription: Text Transcription: x_1 = 3, x_2 = 0, x_3 = -1, x_4 = 5 Ax = b Ax = 0 x_1 = 5r - 2s, x_2 = s, x_3 = s + t, x_4 = t Ax = 0 Ax = b

Suppose that \(x_{1}=-1, x_{2}=2, x_{3}=4, x_{4}=-3\) is a solution of a nonhomogeneous linear system \(A \mathbf{x}=\mathbf{b}\) and that the solution set of the homogeneous system \(A x=0\) is given by the formulas \(x_{1}=-3 r+4 s, \quad x_{2}=r-s, \quad x_{3}=r, \quad x_{4}=s\) a. Find a vector form of the general solution of \(A \mathrm{x}=0\). b. Find a vector form of the general solution of \(A \mathbf{x}=\mathbf{b}\). Equation Transcription: Text Transcription: x_{1} = -1, x_2 = 2, x_3 = 4, x_4 = -3 Ax = b Ax = 0 x_1 = -3r + 4s, x_2 = r - s, x_3 = r, x_4 = s Ax = 0 Ax = b

In Exercises 7-8, find the vector form of the general solution of the linear system $A \mathbf{x}=\mathbf{b}$, and then use that result to find the vector form of the general solution of $A \mathbf{x}=\mathbf{0}$. a. \(\begin{aligned} x_{1}-3 x_{2} &=1 \\ 2 x_{1}-6 x_{2} &=2 \end{aligned}\) b. \(\begin{aligned} x_{1}+x_{2}+2 x_{3}=& 5 \\ x_{1}+x_{3}=&-2 \\ 2 x_{1}+x_{2}+3 x_{3}=& 3 \end{aligned}\) Equation Transcription: Text Transcription: x_1 - 3x_2 = 1 2x_1 - 6x_2 = 2 x_1 + x_2 + 2x_3 = 5 x_1 + x_3 = -2 2x_1 + x_2 + 3x_3 = 3

In Exercises 7-8, find the vector form of the general solution of the linear system $A \mathbf{x}=\mathbf{b}$, and then use that result to find the vector form of the general solution of $A \mathbf{x}=\mathbf{0}$. a. \(\begin{aligned} x_{1}-2 x_{2}+x_{3}+2 x_{4} &=-1 \\ 2 x_{1}-4 x_{2}+2 x_{3}+4 x_{4} &=-2 \\ -x_{1}+2 x_{2}-x_{3}-2 x_{4} &=1 \\ 3 x_{1}-6 x_{2}+3 x_{3}+6 x_{4} &=-3 \end{aligned}\) b. \(\begin{array}{rr} x_{1}+2 x_{2}-3 x_{3}+x_{4} & =4 \\ -2 x_{1}+x_{2}+2 x_{3}+x_{4} & =-1 \\ -x_{1}+3 x_{2}-x_{3}+2 x_{4} & =3 \\ 4 x_{1}-7 x_{2} & -5 x_{4} & =-5 \end{array}\) Equation Transcription: Text Transcription: x_1 - 2x_2 + x_3 + 2x_4 = -1 2x_1 - 4x_2 + 2x_3 + 4x_4 = -2 -x_1 + 2x_2 - x_3 -2x_4 = 1 3x_1 - 6x_2 + 3x_3 + 6x_4 = -3 x_1 + 2x_2 - 3x_3 + x_4 = 4 -2x_1 + x_2 + 2x_3 + x_4 = -1 -x_1 + 3x_2 - x_3 + 2x_4 = 3 4 x_1 - 7x_2 -5x_4 = -5

In Exercises 9-10, find bases for the null space and row space of \(A\). a. \(A=\left[\begin{array}{rrr}1 & -1 & 3 \\ 5 & -4 & -4 \\ 7 & -6 & 2\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 4 & 0 & -2 \\ 0 & 0 & 0\end{array}\right]\) Equation Transcription: = [] = [] Text Transcription: A A = [1 & -1 & 3 \\ 5 & -4 & -4 \\ 7 & -6 & 2] A = [2 & 0 & -1 \\ 4 & 0 & -2 \\ 0 & 0 & 0]

In Exercises 9–10, find bases for the null space and row space of A. a. \(A=\left[\begin{array}{rrrr}1 & 4 & 5 & 2 \\ 2 & 1 & 3 & 0 \\ -1 & 3 & 2 & 2\end{array}\right]\) b. \(A=\left[\begin{array}{rrrrr}1 & 4 & 5 & 6 & 9 \\ 3 & -2 & 1 & 4 & -1 \\ -1 & 0 & -1 & -2 & -1 \\ 2 & 3 & 5 & 7 & 8\end{array}\right]\) Equation Transcription: = [ ] [ ] Text Transcription: A A = [1 & 4 & 5 & 2 \\ 2 & 1 & 3 & 0 \\ -1 & 3 & 2 & 2] A = [1 & 4 & 5 & 6 & 9 \\ 3 & -2 & 1 & 4 & -1 \\ -1 & 0 & -1 & -2 & -1 \\ 2 & 3 & 5 & 7 & 8]

In Exercises 11–12, a matrix in row echelon form is given. By inspection, find a basis for the row space and for the column space of that matrix. a. \(\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & -3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]\) Equation Transcription: [ ] [] Text Transcription: a. [1 & 0 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0] b. [1 & -3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0]

In Exercises 11–12, a matrix in row echelon form is given. By inspection, find a basis for the row space and for the column space of that matrix. a. \(\left[\begin{array}{rrrr}1 & 2 & 4 & 5 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]\) b. \(\left[\begin{array}{rrrr}1 & 2 & -1 & 5 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 1 & -7 \\ 0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [1 & 2 & 4 & 5 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0] [1 & 2 & -1 & 5 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 1 & -7 \\ 0 & 0 & 0 & 1]

a. Use the methods of Examples 6 and 7 to find bases for the row space and column space of the matrix \(A=\left[\begin{array}{rrrrr}1 & -2 & 5 & 0 & 3 \\-2 & 5 & -7 & 0 & -6 \\-1 & 3 & -2 & 1 & -3 \\-3 & 8& -9 & 1 & -9\end{array}\right]\) b. Use the method of Example 9 to find a basis for the row space of \(A\) that consists entirely of row vectors of \(A\). Equation Transcription: [] Text Transcription: A=[_ -3 8 -9 1 -9 ^-1 3 -2 1 -3 ^-2 5 -7 0 -6 ^1 -2 5 0 3] A A

In Exercises 14–15, find a basis for the subspace of \(\boldsymbol{R}^{4}\) that is spanned by the given vectors. \((1, 1, ?4, ?3), (2, 0, 2, ?2), (2, ?1, 3, 2)\) Equation Transcription: ????4 Text Transcription: R^4 (1, 1, ?4, ?3), (2, 0, 2, ?2), (2, ?1, 3, 2)

In Exercises 14–15, find a basis for the subspace of \(\boldsymbol{R}^{4}\) that is spanned by the given vectors. \((1,1,0,0), (0,0,1,1), (-2,0,2,2), (0,-3,0,3)\) Equation Transcription: ????4 Text Transcription: R^4 (1,1,0,0), (0,0,1,1), (-2,0,2,2), (0,-3,0,3)

In Exercises 16–17, find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors. \(\mathbf{v}_{1}=(1,0,1,1), \mathbf{v}_{2}=(-3,3,7,1)\), \(\mathbf{v}_{3}=(-1,3,9,3), \mathbf{v}_{4}=(-5,3,5,-1)\) Equation Transcription: ????1????2 ????3????4 Text Transcription: v_1 = (1, 0, 1, 1), v_2 = (?3, 3, 7, 1), v_3 = (?1, 3, 9, 3), v_4 = (?5, 3, 5, ?1)

In Exercises 16–17, find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors. \(\mathbf{v}_{1}=\left(1,-1,5,2 \mathbf{v}_{2}=(-2,3,1,0)\right.\) \(\mathbf{v}_{3}=(4,-5,9,4), \mathbf{v}_{4}=(0,4,2,-3)\), \(\mathbf{v}_{5}=(-7,18,2,-8)\) Equation Transcription: ????1????2 ????3????4 ????5 Text Transcription: v_1=(1,-1,5,2 v_2=(-2,3,1,0) v_3=(4,-5,9,4), v_4=(0,4,2,-3) v_5=(-7,18,2,-8)

In Exercises 18–19, find a basis for the row space of \(A\) that consists entirely of row vectors of \(A\). The matrix in Exercise 10(a). Equation Transcription: Text Transcription: A A

In Exercises 18–19, find a basis for the row space of \(A\) that consists entirely of row vectors of \(A\). The matrix in Exercise 10(b). Equation Transcription: Text Transcription: A A

Construct a matrix whose null space consists of all linear combinations of the vectors \(\mathbf{v}_{1}=\left[\begin{array}{r}1 \\-1 \\3 \\2\end{array}\right]\) and \(\mathbf{v}_{2}=\left[\begin{array}{r}2 \\0 \\-2 \\4\end{array}\right]\) Equation Transcription: ????1=[] ????2=[] Text Transcription: v_1=[ _2 ^3 ^-1 ^1] v_2=[ _4 ^-2 ^0 ^2]

In each part, let \(A=\left[\begin{array}{rrr}1 & 2 & 0 \\1 & -1 & 4\end{array}\right]\) For the given vector \(\mathbf{b\), find the general form of all vectors \(\mathbf{x}\) in \(R^{3}\) for which \(T_{A}(\mathbf{x})=\mathbf{b}\) if such vectors exist. a. \(\mathbf{b}=(0,0)\) b. \(\mathbf{b}=(1,3)\) c. \(\mathbf{b}=(-1,1)\) Equation Transcription: [] ???? ???? (????)=???? ???? ???? ???? Text Transcription: A=[_1 -1 4 ^1 2 0] b x R^3 TA(x)=b b=(0,0) b=(1,3) b=(-1,1)

In each part, let \(A=\left[\begin{array}{ll}2 & 0 \\0 & 1 \\1 & 1 \\2 & 0\end{array}\right]\) For the given vector \(\mathbf{b\), find the general form of all vectors \(\mathbf{x}\) in \(R^{2}\) for which \(T_{A}(\mathbf{x})=\mathbf{b}\) if such vectors exist. a. \(\mathbf{b}=(0,0,0,0)\) b. \(\mathbf{b}=(1,1,-1,-1)\) c. \(\mathbf{b}=(2,0,0,2)\) Equation Transcription: [] ???? ???? (????)=???? ???? ???? ???? Text Transcription: A=[ _2 0 ^1 1 ^0 1 2 ^0] b x R^3 TA(x)=b b=(0,0,0,0) b=(1,1,-1,-1) b=(2,0,0,2)

Suppose that the characteristic polynomial of some matrix ???? is found to be p(????) = (???? ? 1)(???? ? 3)2(???? ? 4)3. In each part, answer the question and explain your reasoning. a. What can you say about the dimensions of the eigenspaces of ????? b. What can you say about the dimensions of the eigenspaces if you know that ???? is diagonalizable? c. If {v1, v2, v3} is a linearly independent set of eigenvectors of ????, all of which correspond to the same eigenvalue of ????, what can you say about that eigenvalue? Equation Transcription: Text Transcription: A p() = ( - 1)(???? ? 3)2(- 4)3 {v1, v2, v3}

Let \(A=\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\) Show that a. \(A\) is diagonalizable if \((a-d)^{2}+4 b c>0 \\\). b. \(A\) is not diagonalizable if \((a-d)^{2}+4 b c<0 \\\). [Hint: See Exercise 29 of Section 5.1.] Equation Transcription: [] Text Transcription: A=[c da b] (a ? d)^2 + 4bc > 0 (a ? d)^2 + 4bc < 0

In the case where the matrix \(A\) in Exercise 28 is diagonalizable, find a matrix \(P\) that diagonalizes \(A\). [Hint: See Exercise 30 of Section 5.1.] Equation Transcription: Text Transcription: A P

Find the standard matrix \(A\) for the given linear operator, and determine whether that matrix is diagonalizable. If diagonalizable, find a matrix \(P\) that diagonalizes \(A\). \(T\left(x_{1}, x_{2}\right)=\left(2 x_{1}-x_{2}, x_{1}+x_{2}\right)\) Equation Transcription: Text Transcription: A P ????(x-1, x_2) = (2x_1 -x_2, x-1 + x_2)

Find the standard matrix \(A\) for the given linear operator, and determine whether that matrix is diagonalizable. If diagonalizable, find a matrix \(P\) that diagonalizes \(A\). \(T\left(\mathrm{x}_{1},\mathrm{x}_{2}\right)=\left(-\mathrm{x}_{2},-\mathrm{x}_{1}\right)\) Equation Transcription: Text Transcription: A P ????(x1, x2) = (-x2, - x1)

Find the standard matrix \(A\) for the given linear operator, and determine whether that matrix is diagonalizable. If diagonalizable, find a matrix \(P\) that diagonalizes \(A\). \(\begin{array}{r}T\left(x_{1}, x_{2}, x_{3}\right)=\left(8 x_{1}+3 x_{2}-4 x_{3},-3 x_{1}+x_{2}+3 x_{3}\right. \\\left.4 x_{1}+3 x_{2}\right)\end{array}\) Equation Transcription: Text Transcription: A P ????(x1, x2, x3) = (8x1 + 3x2- 4x3, -3x1+ x2 + 3x3, 4x1 + 3x2 )

Find the standard matrix \(A\) for the given linear operator, and determine whether that matrix is diagonalizable. If diagonalizable, find a matrix \(P\) that diagonalizes \(A\). \(T\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(3 \mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{1}-\mathrm{x}_{2}\right)\) ????(x1, x2, x3) = (3x1, x2, x1 ? x2) Equation Transcription: Text Transcription: A P ????(x1, x2, x3) = (3x1 + x2- 4x3, -3 x1+ x2 + 3x3)

If \(P\) is a fixed \(n \times n \\\) matrix, then the similarity transformation \(A \rightarrow P^{-1} A P\) can be viewed as an operator \(S_{P}(A)=P^{-1} A P \\\) on the vector space \(M_{n n}\) of \(n \times n \\\) matrices. a. Show that \(S_{P}\) is a linear operator. b. Find the kernel of \(S_{P}\). c. Find the rank of \(S_{P}\). Equation Transcription: Text Transcription: A P nxn A right arrow P^-1AP S_P(????) = P^-1???????? M_nn S_P

Prove that similar matrices have the same rank and nullity.

Prove that similar matrices have the same trace.

Prove that if ???? is diagonalizable, then so is ????k for every positive integer k. Equation Transcription: Text Transcription: A A^k k

We know from Table 1 that similar matrices, \(A\) and \(B\), have the same eigenvalues. However, it is not true that those eigenvalues have the same corresponding eigenvectors for the two matrices. Prove that if \(B=P^{-1} A P \\\), and \(v\) is an eigenvector of \(B\) corresponding to the eigenvalue \(\lambda\), then \(Pv\) is the eigenvector of \(A\) corresponding to \(\lambda\). Equation Transcription: Text Transcription: A B B=P^-1AP v lambda Pv

Let \(A\) be an \(n \times n \\\) matrix, and let q(????) be the matrix \(\mathrm{q}(A)=\mathrm{a}_{\mathrm{n}} A^{\mathrm{n}}+\mathrm{a}_{\mathrm{n}-1} A^{\mathrm{n}-1}+\cdots+\mathrm{a}_{1} A+\mathrm{a}_{0} I_{\mathrm{n}} \\\) a. Prove that if \(B=P^{-1} A P \\\), then \(q(B)=P^{-1} q(A) P \\\). b. Prove that if \(A\) is diagonalizable, then so is \(q(A) \\\). Equation Transcription: Text Transcription: A nxn q(????) = anAn + an-1 An-1 + + a1???? + a0 In B=P^-1AP q(B)=P^-1 q(A)P q(A)

Prove that if ???? is a diagonalizable matrix, then the rank of ???? is the number of nonzero eigenvalues of ????.

This problem will lead you through a proof of the fact that the algebraic multiplicity of an eigenvalue of an \(n \times n\) matrix \(A\) is greater than or equal to the geometric multiplicity. For this purpose, assume that \(\lambda_{0} \\\) is an eigenvalue with geometric multiplicity \(k\). a. Prove that there is a basis \(B=\left\{u_{1}, u_{2}, \ldots, u_{n}\right\} \\\) for \(R^{n}\) in which the first \(k\) vectors of \(B\) form a basis for the eigenspace corresponding to \(\lambda_{0} \\\). b. Let\(P\) be the matrix having the vectors in \(B\) as columns. Prove that the product \(AP\) can be expressed as \(A P=P\left[\begin{array}{cc}\lambda_{0} I_{k} & X \\0 & Y\end{array}\right]\) [Hint: Compare the first \(k\) column vectors on both sides.] c. Use the result in part (b) to prove that \(A\) is similar to \(C=\left[\begin{array}{cc}\lambda_{0} I_{k} & X \\0 & Y\end{array}\right]\) and hence that \(A\) and \(C\) have the same characteristic polynomial. d. By considering det \((\lambda I-C) \\\), prove that the characteristic polynomial of \(C\) (and hence \(A\)) contains the factor \((\lambda-\lambda 0)\) at least \(k\) times, thereby proving that the algebraic multiplicity of \(\lambda_{0} \\\).is greater than or equal to the geometric multiplicity \(k\). Equation Transcription: [] [] Text Transcription: n x n A lambda_0 k B = {u_1, u_2, . . . , u_n} R^n AP=P [ 0 Y lambda_0 I_k X ] C= [ 0 Y lambda_0 I_k X ] (lambda ???? -????) ( lambda? lambda 0)

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. An \(n \times n \\\) matrix with fewer than \(n\\) distinct eigenvalues is not diagonalizable. Equation Transcription: Text Transcription: n nxn

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. An \(n \times n \\\)matrix with fewer than \(n\) linearly independent eigenvectors is not diagonalizable. Equation Transcription: Text Transcription: n nxn

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If \(A\) and \(B\) are similar \(n \times n \\\) matrices, then there exists an invertible \(n \times n \\\) matrix ???? such that \(PA=BP\). Equation Transcription: Text Transcription: A nxn P PA=BP

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If \(A\) is diagonalizable, then there is a unique matrix \(P\) such that \(P^{-1} A P \\\) is diagonal. Equation Transcription: Text Transcription: A P P^-1AP

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If \(A\) is diagonalizable and invertible, then \(A^{-1}\) is diagonalizable. Equation Transcription: Text Transcription: A A^-1

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If \(A\) is diagonalizable, then \(A^{T}\) is diagonalizable. Equation Transcription: Text Transcription: A A^T

In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If there is a basis for \(R^{n}\)consisting of eigenvectors of an \(n \times n\) matrix \(A\), then \(A\) is diagonalizable. Equation Transcription: Text Transcription: Nxn R^n A

TF. In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If every eigenvalue of a matrix ???? has algebraic multiplicity 1, then ???? is diagonalizable.

TF. In parts (a)–(i) determine whether the statement is true or false, and justify your answer. If 0 is an eigenvalue of a matrix \(4 \times 4 \\\), then \(A^{2}\) is singular. Equation Transcription: Text Transcription: 0 A^2 A

Generate a random \(4 \times 4 \\\) matrix \(A\) and an invertible \(4 \times 4 \\\) matrix \(P\) and then confirm, as stated in Table 1, that \(P^{-1} A P \\\) and ???? have the same a. determinant. b. rank. c. nullity. d. trace. e. characteristic polynomial. f . eigenvalues. Equation Transcription: Text Transcription: 4x4 A P^-1AP P

a. Use Theorem 5.2.1 to show that the following matrix is diagonalizable. \(A=\left[\begin{array}{rrr}-13 & -60 & -60 \\10 & 42 & 40 \\-5 & -20 & -18\end{array}\right]\) b. Find a matrix \(P\) that diagonalizes \(A\). c. Use the method of Example 6 to compute \(A^{10}\), and check your result by computing \(A^{10}\) directly. Equation Transcription: [] Text Transcription: A =[ -5 -20 -18 10 42 40 -13 -6 0 -60] P A A^10

Use Theorem 5.2.1 to show that the following matrix is not diagonalizable. \(A=\left[\begin{array}{rrr}-10 & 11 & -6 \\-15 & 16 & -10 \\-3 & 3 & -2\end{array}\right]\) Equation Transcription: [] Text Transcription: A =[ -3 -3 -2 -15 16 -10 -0 1 1 -6]

Find \(\bar{u}, \operatorname{Re}(u), \operatorname{Im}(u) \\\), and \(\|u\|\). \(u=(2-i, 4 i, 1+i)\) Equation Transcription: Text Transcription: U,Re?(u),Im?(u) ||u|| u=(2-i,4i,1+i)

Find \(\bar{u}, \operatorname{Re}(u), \operatorname{Im}(u) \\\), and \(\|u\|\). \(u=(6,1+4 i, 6-2 i)\) Equation Transcription: Text Transcription: u,Re?(u),Im?(u) ||u|| u=(6,1+4i,6-2i)

a. Show that if \(0<\theta<\pi \\\), then \(A=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\\sin \theta & \cos \theta\end{array}\right]\end{aligned}\) has no real eigenvalues and consequently no real eigenvectors b. Give a geometric explanation of the result in part (a). Equation Transcription: [] Text Transcription: 0< theta < pi A = [sin theta cos theta cos theta -sin theta ]

Find the eigenvalues of \(A=\left[\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\k^{3} & -3 k^{2} & 3 k\end{array}\right]\) Equation Transcription: [] Text Transcription: A = [k^3 -3k^2 3k 0 0 1 0 1 0 ]

a. Show that if \(D\) is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix \(S\) such that \(S^{2}=D \\\). b. Show that if \(A\) is a diagonalizable matrix with nonnegative eigenvalues, then there is a matrix \(S\) such that \(S^{2}=A\). c. Find a matrix \(S\) such that \(S^{2}=A\), given that \(A=\left[\begin{array}{lll}1 & 3 & 1 \\0 & 4 & 5 \\0 & 0 & 9\end{array}\right]\) Equation Transcription: [] Text Transcription: D S S^2=D S^2=A

Given that \(A\) and \(B\) are similar matrices, in each part determine whether the given matrices are also similar. a. \(A^{T} \\\) and \(B^{T} \\\) b. \(A^{k} \\\) and \(B^{k} \\\) (\(k\) is a positive integer) c. \(A^{-1} \\\) and \(B^{-1} \\\) (if \(A\) is invertible) Equation Transcription: Text Transcription: A B A^T B^T A^k B^k k A^-1 B^-1

Prove: If \(A\) is a square matrix and \(p(\lambda)=\operatorname{det}(\lambda I-A) \\\) is the characteristic polynomial of \(A\), then the coefficient of \(\lambda^{n-1} \\\) in \(p(\lambda)\) is the negative of the trace of \(A\). Equation Transcription: Text Transcription: A p(lambda)=det?(lambda I-A) lambda^n-1 p(lambda)

Prove: If \(b \neq 0 \\\), then \(\qquad A=\left[\begin{array}{ll}a & b \\0 & a\end{array}\right]\) is not diagonalizable. Equation Transcription: [] Text Transcription: b not = 0 A =[0 a a b]

In advanced linear algebra, one proves the Cayley–Hamilton Theorem, which states that a square matrix \(A\) satisfies its characteristic equation; that is, if \(c_{0}+c_{1} \lambda+c_{2} \lambda^{2}+\cdots+c_{n-1} \lambda^{n-1}+\lambda^{n}=0 \\\) is the characteristic equation of \(A\), then \(c_{0} I+c_{1} A+c_{2} A^{2}+\cdots+c_{n-1} A^{n-1}+A^{n}=0 \\\) Verify this result for a. \(A=\left[\begin{array}{ll}3 & 6 \\1 & 2\end{array}\right]\) B. \(A=\left[\begin{array}{rrr}0 & 1 & 0 \\0 & 0 & 1 \\1 & -3 & 3\end{array}\right]\) Equation Transcription: [] [] Text Transcription: C_0+c_1+c_ 2 lambda^2 +?+c_n-1 lambda^n-1 + lambda ^n=0 c_0I+c_1 A+ c_2 A^2+?+c_n-1 A^n-1+ A^n=0 A =[1 2 3 6] A =[1 -3 3 0 0 1 0 1 0 ]

Use the Cayley–Hamilton Theorem, stated in Exercise 7. a. Use Exercise 28 of Section 5.1 to establish the Cayley– Hamilton Theorem for \(2x2\) matrices. b. Prove the Cayley–Hamilton Theorem for \(nxn\) diagonalizable matrices. Equation Transcription: Text Transcription: 2x2 nxn

The Cayley–Hamilton Theorem provides a method for calculating powers of a matrix. For example, if \(A\) is a \(2 \times 2 \\\) matrix with characteristic equation \(c_{0}+c_{1} \lambda+\lambda^{3}=0 \\\) The \(c_{0} I+c_{1} A+A^{2}=0\) so \(A^{2} &=-c_{1} A-c_{0} I \\\) Multiplying through by \(A\) yields \(A^{3}=-c_{1} A^{2}-c_{0} A \\\), which expresses \(A^{3} \\\) in terms of \(A^{2} \\\) and \(A\), and multiplying through by \(A^{2} \\\) yields \(A^{4}=-c_{1} A^{3}-c_{0} A^{2}\), which expresses \(A^{4} \\\) in terms of \(A^{3} \\\) and \(A^{2} \\\). Continuing in this way, we can calculate successive powers of by expressing them in terms of lower powers. Use this procedure to calculate \(A^{2} \\\), \(A^{3} \\\),\(A^{4} \\\), and \(A^{5} \\\) for \(A &=\left[\begin{array}{ll}3 & 6 \\1 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: A 2x2 c0+c1+3=0 c0I+c1A+A2= 0, A2=-c1A-c0I A3=-c1A2-c0A A3 A2 A5 A4=-c1A3-c0A2 A =[1 2 3 6]

Use the method of the preceding exercise to calculate \(A^{3} \\\) and \(A^{4} \\\) for \(A=\left[\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\1 & -3 & 3\end{array}\right]\) Equation Transcription: [] Text Transcription: A^3 A^4 A =[1 -3 3 0 0 1 0 1 0]

Find the eigenvalues of the matrix \(A=\left[\begin{array}{cccc}c_{1} & c_{2} & \cdots & c_{n} \\c_{1} & c_{2} & \cdots & c_{n} \\\vdots & \vdots & & \vdots \\c_{1} & c_{2} & \cdots & c_{n}\end{array}\right]\) Equation Transcription: [] Text Transcription: [c_1 c_2 . . . c_n. . . . . . . . . c_1 c_2 . . . c_n c_1 c_2 . . . c_n]

a. It was shown in Exercise 37 of Section that if \(A\) is an \(n \times n \\\) matrix, then the coefficient of \)\lambda^{n} \\\) in the characteristic polynomial of \(A\) is (A polynomial with this property is called monic.) Show that the matrix \(\left[\begin{array}{cccccc}0 & 0 & 0 & \cdots & 0 & -c_{0} \\1 & 0 & 0 & \cdots & 0 & -c_{1} \\0 & 1 & 0 & \cdots & 0 & -c_{2} \\\vdots & \vdots & \vdots & & \vdots & \vdots \\0 & 0 & 0 & \cdots & 1 & -c_{n-1}\end{array}\right]\) has characteristic polynomial \(p(\lambda)=c_{0}+c_{1} \lambda+\cdots+c_{s-1} \lambda^{n-1}+\lambda^{n} \\\) This shows that every monic polynomial is the characteristic polynomial of some matrix. The matrix in this example is called the companion matrix of \(p(\lambda) \\\). [Hint: Evaluate all determinants in the problem by adding a multiple of the second row to the first to introduce a zero at the top of the first column, and then expanding by cofactors along the first column. b. Find a matrix with characteristic polynomial \(p(\lambda)=1-2 \lambda+\lambda^{2}+3 \lambda^{3}+\lambda^{4}\) Equation Transcription: [] Text Transcription: A lambda^n nxn [0 0 0 . . . 1 -c_n-1. . . . .. . . . .. . . . .0 1 0 . . . 0 -c_2 1 0 0 . . . 0 -c_1 0 0 0 . . . 0 -c_0] p(lambda)=c_0 + c_1 lambda +?+c_s-1 lambda^n-1 + lambda^n p(lambda) p(lambda)=1-2lambda+ lambda^2+3 lambda^3 + lambda^4

A square matrix \(A\) is called nilpotent if \(A^{n}=0\) for some positive integer \(n\). What can you say about the eigenvalues of a nilpotent matrix? Equation Transcription: Text Transcription: A A^n=0 n

Prove: If \(A\) is an \(n \times n \\\) matrix with real entries and \(n\) is odd, then \(A\) has at least one real eigenvalue. Equation Transcription: Text Transcription: nxn A n

Find a \(3 \times 3 \\\) matrix \(A\) that has eigenvalues \(\lambda=0,1\), and with corresponding eigenvectors \(\left[\begin{array}{r}0 \\1 \\-1\end{array}\right], \quad\left[\begin{array}{r}1 \\-1\\1\end{array}\right], \quad\left[\begin{array}{l}0 \\1 \\1\end{array}\right]\) respectively. Equation Transcription: [], [], [] Text Transcription: 3x3 A lambda=0,1 [-1 1 0 ], [ 1 -1 1 ], [ 1 10 ]

Suppose that a \(4 \times 4\) matrix \(A\) has eigenvalues \(\lambda_{1}=1\), \lambda_{2}=-2, \lambda_{3}=3\), and \(\lambda_{4}=-3\) a. Use the method of Exercise 24 of Section 5.1 to find det \(A\). b. Use Exercise 5 above to find tr \(A\). Equation Transcription: Text Transcription: 4 x 4 A lamba_1=1, lamba_2=-2, lamba_3=3, lamba_ 4=-3 A A

Lot \(A\) be a square matrix such that \(A^{3}=A\). What ean gou Ray about the eigenvalues of \(A\)? Equation Transcription: Text Transcription: A A^3=A A

a. Solve the system \(y_{1}^{\prime}=y_{1}+3 y_{2}\) \(y_{2}^{\prime}=2 y_{1}+4 y_{2}\) b. Find the solution satisfying the initial conditions \(y_{1}(0)=5\) and y_{2}(0)=6\). Equation Transcription: Text Transcription: y'_1=y_1+3y_2 y'_2=2y_1+4y_2 y_1(0)=5 y_2(0)=6

Let \(A\) be a \(3 \times 3\) matrix, one of whose eigenvalues is Given that both the sum and the product of all three eigenvalues is 6, what are the possible values for the remaining two eigenvalues? Equation Transcription: Text Transcription: A 3 x 3

Show that the matrices \(A=\left[\begin{array}{lll}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{array}\right]\) and \(D=\left[\begin{array}{ccc}d_{1} & 0 & 0 \\0 & d_{2} & 0 \\0 & 0 & d_{3}\end{array}\right]\) are similar if \(d_{k}=\cos \frac{2 \pi k}{3}+i \sin \frac{2 \pi k}{3}(k=1,2,3)\) Equation Transcription: [] [] Text Transcription: A=[ _1 0 0 ^0 0 1 ^0 1 0] D=[ _0 0 d_3 ^0 d^2 0 ^d_1 0 0] d_k=cos 2 pi k/3+i sin 2 pi k/3(k=1,2,3)