?Let be the subspace of \(P_{3}\) spanned by the vectors\(p_{1}=1+5 x-3 x^{2}-11 x^{3}

Use the method of Example 3 to show that the following set of vectors forms a basis for \(\{(2,1),(3,0)\}\) Equation Transcription: Text Transcription: {(2, 1), (3, 0)}

Use the method of Example 3 to show that the following set of vectors forms a basis for \(R^{3}\). \(\{(3,1,-4),(2,5,6),(1,4,8)\}\) Equation Transcription: Text Transcription: R^3 {(3, 1, -4), (2, 5, 6), (1, 4, 8)

Show that the following polynomials form a basis for \(P_{2}\). \(x^{2}+1, x^{2}-1,2 x-1\) Equation Transcription: Text Transcription: P_2 x^2 +1, x^2 -1, 2x-1

Show that the following polynomials form a basis for \(P_{3}\). \(1+x, 1-x, 1-x^{2}, 1-x^{3}\) Equation Transcription: Text Transcription: P_3 1+x, 1-x, 1-x^2 , 1-x^3

Show that the following matrices form a basis for \(M_{22}\). \(\left[\begin{array}{cc} 3 & 6 \\ 3 & -6 \end{array}\right],\left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right],\left[\begin{array}{cc} 0 & -8 \\ -12 & -4 \end{array}\right],\left[\begin{array}{cc} 1 & 0 \\ -1 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: M_22 [3 6 _ 3 -6 ], [0 -1 _ -1 0 ], [0 -8 _ -12 -4 ], [1 0 _ -1 2 ]

Show that the following matrices form a basis for \(M_{22}\). \(\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right],\left[\begin{array}{rr} 1 & -1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{rc} 0 & -1 \\ -1 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: M_22 [1 1 _ 1 1], [1 -1 _ 0 0 ], [0 -1 _ 1 0 ], [1 0 _ 0 0 ]

In each part, show that the set of vectors is not a basis for \(R^{3}\). \(a.\{(2,-3,1),(4,1,1),(0,-7,1)\}\) \(b. \{(1,6,4),(2,4,-1),(-1,2,5)\}\) Equation Transcription: Text Transcription: R^3 {(2, -3, 1), (4, 1, 1), (0, -7, 1)} {(1, 6, 4), (2, 4, -1), (-1, 2, 5)}

Show that the following vectors do not form a basis for \(P_{2}\). \(1-3 x+2 x^{2}, 1+x+4 x^{2}, 1-7 x\) Equation Transcription: Text Transcription: P_2 1-3x+2x^2 , 1+x+4x^2 , 1-7x

Show that the following matrices do not form a basis for \(M_{22}\). \(\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}\right],\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right],\left[\begin{array}{rr} 1 & -1 \\ 1 & 0 \end{array}\right],\left[\begin{array}{rr} 0 & -1 \\ 1 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: M_22 [1 0 _ 1 1 ], [2 -2 _ 3 2 ], [1 -1 _ 1 0 ], [0 -1 _ 1 1 ]

Let be the space spanned by \(v_{1}=\cos ^{2} x, v_{2}=\sin ^{2} x, v_{3}=\cos 2 x\) a. Show that \(S=\left\{v_{1}, v_{2}, v_{3}\right\}\) is not a basis for . b. Find a basis for . Equation Transcription: Text Transcription: v_1 =cos^2 x, v_2 =sin^2 x, v_3 =cos 2x S={v_1 , v_2 , v_3 }

Find the coordinate vector of relative to the basis \(S=\left\{u_{-} 1, u_{-} 2\right\}\) for \(R^{2}\) \(\text { a. } u_{1}=(2,-4), u_{2}=(3,8) ; w=(1,1)\) \(\text { b. } u_{1}=(1,1), u_{2}=(0,2) ; w=(a, b)\) Equation Transcription: Text Transcription: S={u_1 , u_2 } R^2 u_1 =(2, -4), u_2 =(3, 8); w=(1, 1) u_1 (1, 1), u_2 =(0, 2); w=(a, b)

Find the coordinate vector of relative to the basis \(S=\left\{u_{-} 1, u_{-} 2\right\}\) for \(R^{2}\) \(\text { a. } u_{1}=(1,-1), u_{2}=(1,1) ; w=(1,0)\) \(\text { b. } u_{1}=(1,-1), u_{2}=(1,1) ; w=(0,1)\) Equation Transcription: Text Transcription: S={u_1 , u_2 } R^2 u_1 =(1, -1), u_2 =(1, 1); w=(1, 0) u_1 =(1, -1), u_2 =(1, 1); w=(0, 1)

Find the coordinate vector of relative to the basis \(S=\left\{v_{1}, v_{2}, v_{3}\right\}\) for \(R^{3}\). \(\text { a. } v=(2,-1,3) ; v_{1}=(1,0,0), v_{2}=(2,2,0), v_{3}=(3,3,3)\) \(\text { b. } v=(5,-12,3) ; v_{1}=(1,2,3), v_{2}=(-4,5,6), v_{3}=(7,-8,9)\) Equation Transcription: Text Transcription: S={v_1 , v_2 , v_3 } R^3 v = (2, -1, 3); v_1 =(1, 0, 0), v_2 =(2, 2, 0), v_3 =(3, 3, 3) v = (5, -12, 3); v_1 =(1, 2, 3), v_2 =(-4, 5, 6), v_3 =(7, -8, 9)

Find the coordinate vector of relative to the basis \(S=\left\{p_{1}, p_{2}, p_{3}\right\}\) for \(P_{2}\) \(\text { a. } p=4-3 x+x^{2} ; p_{1}=1, p_{2}=x, p_{3}=x^{2}\) \(\text { b. } p=2-x+x^{2} ; p_{1}=1, p_{2}=1+x^{2}, p_{3}=x+x^{2}\) Equation Transcription: Text Transcription: S={p_1 , p_2 , p_3 } P_2 p=4-3x+x^2 ; p_1 =1, p_2 , p_3=x^2 p=2-x+x^2 ;p_1 =1, p_2 =1+x^2 , p_3 =x+x^2

In Exercises 15-16, first show that the set \(S=\left\{A_{1}, A_{2}, A_{3}, A_{4}\right\}\) is a then express as a linear combination of the in , and then find the coordinate vector of relative to . \(A_{1}=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right], A_{2}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right], A_{3}=\left[\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right]\) \(A_{4}=\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right], A=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: S={A_1 , A_2 , A_3 , A_4 } A_1 =[1 1 _ 1 1], A_2 =[0 1 _ 1 1], A_3 =[0 0 _ 1 1] A_4 =[0 0 _ 0 1], A =[1 0 _ 1 0]

In Exercises 15-16, first show that the set \(S=\left\{A_{1}, A_{2}, A_{3}, A_{4}\right\}\) is a then express as a linear combination of the in , and then find the coordinate vector of relative to . \(A_{1}=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right], A_{2}=\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right], A_{3}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\) \(A_{4}=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right], A=\left[\begin{array}{ll} 6 & 2 \\ 5 & 3 \end{array}\right]\) Equation Transcription: Text Transcription: S={A_1 , A_2 , A_3 , A_4 } A_1 =[1 0 _ 1 0], A_2 =[1 1 _ 0 0], A_3 =[1 0 _ 0 1] A_4 =[0 0 _ 1 0], A =[6 2 _ 5 3]

In Exercises , first show that the set \(S=\left\{p_{1}, p_{2}, p_{3}\right\}\) is a basis for \(p_{2}\), then express as a linear combination of the vectors in , and then find the coordinate vector of p relative to . \(p_{1}=1+x+x^{2}, p_{2}=x+x^{2}, p_{3}=x^{2}\) \(p=7-x+2 x^{2}\) Equation Transcription: Text Transcription: S={p_1 , p_2 , p_3 } p_2 p_1 =1+x+x^2 , p_2 =x+x^2 , p_3 =x^2 p=7-x+2x^2

In Exercises , first show that the set \(S=\left\{p_{1}, p_{2}, p_{3}\right\}\) is a basis for \(p_{2}\), then express as a linear combination of the vectors in , and then find the coordinate vector of p relative to \(p_{1}=1+2 x+x^{2}, p_{2}=2+9 x, p_{3}=3+3 x+4 x^{2}\) \(p=2+17 x-3 x^{2}\) Equation Transcription: Text Transcription: S={p_1 , p_2 , p_3 } p_2 p_1 =1+2x+x^2 , p_2 =2+9x, p_3 =3+3x+4x^2 p=2+17x-3x^2

In words, explain why the sets of vectors in parts (a) to (d) are not bases for the indicated vector spaces. \(a. u_{1}=(1,2), u_{2}=(0,3), u_{3}=(1,5) \text { for } R^{2}\) \(b. u_{1}=(-1,3,2), u_{2}=(6,1,1) \text { for } R^{3}\) \(c. p_{1}=1+x+x^{2}, p_{2}=x \text { for } P_{2}\) \(d. A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 3 \end{array}\right], B=\left[\begin{array}{rr} 6 & 0 \\ -1 & 4 \end{array}\right], C=\left[\begin{array}{ll} 3 & 0 \\ 1 & 7 \end{array}\right], D=\left[\begin{array}{ll} 5 & 0 \\ 4 & 2 \end{array}\right] \text { for } M_{22}\) Equation Transcription: Text Transcription: u_1 =(1, 2), u_2 =(0, 3), u_3 =(1, 5) for R^2 u_1 =(-1, 3, 2), u_2 =(6, 1, 1) for R^3 p_1 =1+x+x^2 , p_2 =x for P_2 A=[1 0 _ 2 3], B[6 0 _ -1 4], C=[3 0 _ 1 7], D=[5 0 _ 4 2]

In any vector space a set that contains the \(\text { Zero }\) vector must be linearly dependent. Explain why this is so. Equation Transcription: Text Transcription: Zero

In each part, let \(T_{A}: R^{3} \rightarrow R^{3}\) be multiplication by , and let \(\left\{e_{1}, e_{2}, e_{3}\right\}\) be the standard basis for \(R^{3}\). Determine whether the set \(\left\{T_{A}\left(e_{1}\right), T_{A}\left(e_{2}\right), T_{A}\left(e_{3}\right)\right\}\) is linearly independent in \(R^{2}\). \(\text { a. } A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & -3 \\ -1 & 2 & 0 \end{array}\right]\) \(b . A=\left[\begin{array}{ccc} 1 & 1 & 2 \\ 0 & 1 & 1 \\ -1 & 2 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T_A : R^3 right arrow R^3 {e_1 , e_2 , e_3} R^3 {T_A (e_1 ), T_A (e_2 ), T_A (e_3 )} R^2 A=[1 1 1 _ 0 1 -3 _ -1 2 0] A=[1 1 2 _ 0 1 1 _ -1 2 1]

In each part, let \(T_{A}: R^{3} \rightarrow R^{3}\) be multiplication by , and let \(u=(1,-2,1)\). Find the coordinate vector of \(T_{A}(u)\) relative to the basis \(S=\{(1,1,0),(0,1,1),(1,1,1)\}\) for \(R^{3}\). \(\text { a. } A=\left[\begin{array}{ccc} 2 & -1 & 0 \\ 1 & 1 & 1 \\ 0 & -1 & 2 \end{array}\right]\) \(b . A=\left[\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: T_A : R^3 right arrow R^3 u=(1, -2, 1) T_A (u) S={(1, 1, 0), (0, 1, 1), (1, 1, 1)} R^3 A=[2 -1 0 _ 1 1 1 _ 0 -1 2] A=[0 1 0 _ 1 0 1 _ 0 0 1]

The accompanying figure shows a rectangular -coordinate system determined by the unit basis vectors and and an \(x^{\prime} y^{\prime}\) -coordinate system determined by unit basis vectors \(u_{1} \text { and } u_{2}\). Find the \(x^{\prime} y^{\prime}\) -coordinates of the points whose -coordinates are given. \(a. (\sqrt{3}, 1)\) \(b. (1,0)\) \(c. (0,1)\) \(d. (a, b)\) Equation Transcription: Text Transcription: x prime y prime u_1 and u_2 (sqrt3, 1) (1, 0) (0, 1) (a, b)

The accompanying figure shows a rectangular -coordinate system and an \(x^{\prime} y^{\prime}\) -coordinate system with skewed axes. Assuming that 1 -unit scales are used on all the axes, find the \(x^{\prime} y^{\prime}\) -coordinates of the points whose -coordinates are given. \(a. (1,1)\) \(b. (1,0)\) \(c. (0,1)\) \(d. (a, b)\) Equation Transcription: Text Transcription: x prime y prime (1, 1) (1, 0) (0, 1) (a, b)

The first four Hermite polynomials [named for the French mathematician Charles Hermite are \(1,2 t,-2+4 t^{2},-12 t+8 t^{2}\) These polynomials have a wide variety of applications in physics and engineering a. Show that the first four Hermite polynomials form a basis for \(P_{3}\). b. Let be the basis in part (a). Find the coordinate vector of the polynomial \(p(t)=-1-4 t+8 t^{2}+8 t^{2}\) relative to B. Equation Transcription: Text Transcription: 1, 2t, -2+4t^2 , -12t+8t^2 P_3 p(t)=-1-4t+8t^2 +8t^2

The first four Laguerre polynomials [named for the French mathematician Edmond Laguerre are \(1,1-t, 2-4 t+t^{2}, 6-18 t+9 t^{2}-t^{3}\) a. Show that the first four Laguerre polynomials form a basis for \(P_{3}\). b. Let be the basis in part (a). Find the coordinate vector of the polynomial \(p(t)=-10 t+9 t^{2}-t^{3}\) relative to Equation Transcription: Text Transcription: 1, 1-t, 2-4t+t^2 , 6-18t+9t^2 -t^3 P_3 p(t)=-10t+9t^2 -t^3

Consider the coordinate vectors \([w]_{s}=\left[\begin{array}{c} 6 \\ -1 \\ 4 \end{array}\right],[q]_{s}=\left[\begin{array}{l} 3 \\ 0 \\ 4 \end{array}\right] [B]_{s}=\left[\begin{array}{c} -8 \\ 7 \\ 6 \\ 3 \end{array}\right]\) a. Find w if S is the basis in Exercise 2. b. Find q if S is the basis in Exercise 3. c. Find B if S is the basis in Exercise 5. Equation Transcription: Text Transcription: [w]_s =[6 _ -1 _ 4], [q]_s =[3 _ 0 _ 4], [B]_s =[-8 _ 7 _ 6 _ 3]

The basis that we gave for \(M_{22}\) in Example 4 consisted of non-invertible matrices. Do you think that there is a basis for \(M_{22}\) consisting of invertible matrices? Justify your answer. Equation Transcription: Text Transcription: M_22

Prove that \(R^{\infty}\) is an infinite-dimensional vector space. Equation Transcription: Text Transcription: R^infinity

Let \(T_{A}: R^{n} \rightarrow R^{n}\) be multiplication by an invertible matrix , and let \(\left\{u_{1}, u_{2}, \ldots, u_{n}\right\}\) be a basis for \(R^{n}\). Prove that \(\left\{T_{A}\left(u_{1}\right), T_{A}\left(u_{2}\right), \ldots, T_{A}\left(u_{n}\right)\right\}\) is also a basis for \(R^{n}\). Equation Transcription: Text Transcription: T_A :R^n right arrow R^n {u_1 , u_2 ,..., u_n } R^n {T_A (u_1 ), T_A (u_2 ),..., T_A (u_n )}

Prove that if \(V\) is a subspace of a vector space \(W\) and if \(V\) is infinite-dimensional, then so is \(W\). Equation Transcription: Text Transcription: V W

TF. In parts (a)-(e) determine whether the statement is true or false, and justify your answer. If \(V=\operatorname{span}\left\{v_{1}, \ldots, v_{n}\right\}\), then \(\left\{v_{1}, \ldots, v_{n}\right\}\) is a basis for . Equation Transcription: Text Transcription: V=span{v_1 ,..., v_n } {v_1 ,..., v_n }

TF. In parts (a)-(e) determine whether the statement is true or false, and justify your answer. Every linearly independent subset of a vector space \(V\) is a basis for \(V\). Equation Transcription: Text Transcription: V

TF. In parts (a)-(e) determine whether the statement is true or false, and justify your answer. If \(\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) is a basis for a vector space \(V\), then every vector in \(V\) can be expressed as a linear combination of \(v_{1}, v_{2}, \ldots, v_{n}\). Equation Transcription: Text Transcription: {v_1 , v_2 ,..., v_n } V v_1 , v_2 ,..., v_n

TF. In parts (a)-(e) determine whether the statement is true or false, and justify your answer. The coordinate vector of a vector \(x \text { in } R^{n}\) relative to the standard basis for \(R^{n} \text { is } x\). Equation Transcription: Text Transcription: x in R^n R^n is x

TF. In parts (a)-(e) determine whether the statement is true or false, and justify your answer. Every basis of \(P_{4}\) contains at least one polynomial of degree 3 or less. Equation Transcription: Text Transcription: P_4

Let be the subspace of \(P_{3}\) spanned by the vectors \(p_{1}=1+5 x-3 x^{2}-11 x^{3}, \quad p_{2}=7+4 x-x^{2}+2 x^{3}\) \(p_{3}=5+x+9 x^{2}+2 x^{3}, \quad p_{4}=3-x+7 x^{2}+5 x^{3}\) a. Find a basis for . b. Find the coordinate vector of \(p=19+18 x-13 x^{2}-10 x^{3}\) relative to the basis you obtained in part (a). Equation Transcription: Text Transcription: P_3 p_1 =1+5x-3x^2 -11x^3, p_2 =7+4x-x^2 +2x^3 p_3 =5+x+9x^2 +2x^3 , p_4 =3-x+7x^2 +5x^3 p=19+18x-13x^2 -10x^3

Let be the subspace of \(C^{\infty}(-\infty, \infty)\) spanned by the vectors in the set \(B=\left\{1, \cos x, \cos ^{2} x, \cos ^{3} x, \cos ^{4} x, \cos ^{5} x\right\}\) and accept without proof that is a basis for . Confirm that the following vectors are in , and find their coordinate vectors relative to . \(f_{0}=1, f_{1}=\cos x, f_{2}=\cos 2 x, f_{3}=\cos 3 x, f_{4}=\cos 4 x, f_{5}=\cos 5 x\) Equation Transcription: Text Transcription: c^infinity (-infinity, infinity) B={1, cosx, cos^2 x, cos^3 x, cos^4 x, cos^5 s} f_0 =1, f_1 =cos x, f_2 =cos 2x, f_3 =cos 3x, f_4 =cos 4x, f_5 =cos 5x