?In the Remark following Example 8 we discussed two alternative ways to perform the

In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on \(R^{2}\) . a. \((0,1),(2,0)\) b. \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right),\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) c. \(\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right),\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) d. \((0,0),(0,1)\) Equation Transcription: ????2 Text Transcription: R^2 (0, 1), (2, 0) (- 1/sqrt 2 , 1/sqrt 2 ), (1/sqrt 2 , 1/sqrt 2 ) (- 1/sqrt 2 , - 1/sqrt 2) , (1/sqrt 2 , 1/sqrt 2 ) (0, 0), (0, 1)

In each part, determine whether the set of vectors is orthogonal and whether it is orthonormal with respect to the Euclidean inner product on \(R^{3}\). a. \(\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right),\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right),\left(-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)\) b. \(\left(\frac{2}{3},-\frac{2}{3}, \frac{1}{3}\right),\left(\frac{2}{3}, \frac{1}{3},-\frac{2}{3}\right),\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)\) c. \((1,0,0),\left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right),(0,0,1)\) d. \(\left(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}}\right),\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}, 0\right)\) Equation Transcription: ????3 Text Transcription: R^3 (1/sqrt 2 ,0, 1/sqrt 2) , (1/sqrt 3 , 1/sqrt 3,-1/sqrt 3) ,(-1/sqrt 2 ,0, 1/sqrt 2 ) (2/3 ,-2/3 , 1/3) , (2/3 , 1/3,-2/3) ,(1/3 ,2/3 , 2/3 ) (1,0,0),(0,1/sqrt 2,1/sqrt 2),(0,0,1) (1/sqrt 6,1/sqrt 6,-2/sqrt 6),(1/sqrt 2 ,-1/sqrt 2,0)

In each part, determine whether the set of vectors is orthogonal with respect to the standard inner product on \(P_{2}\) (see Example 7 of Section 6.1). a. \(p_{1}(x)=\frac{2}{3}-\frac{2}{3} x+\frac{1}{3} x^{2}, p_{2}(x)=\frac{2}{3}+\frac{1}{3} x-\frac{2}{3} x^{2}\), \(p_{3}(x)=\frac{1}{3}+\frac{2}{3} x+\frac{2}{3} x^{2}\) b. \(p_{1}(x)=1, p_{2}(x)=\frac{1}{\sqrt{2}} x+\frac{1}{\sqrt{2}} x^{2}, p_{3}(x)=x^{2}\) Equation Transcription: ????2 Text Transcription: P_2 p_1(x)=2/3-2/3x+1/3x^2, p_2(x)=2/3+1/3x-2/3x^2, p_3(x)=1/3+2/3x+2/3x^2 p_1(x)=1, p_2(x)=1/sqrt 2x+1/sqrt 2 x^2, p_3(x)=x^2

In each part, determine whether the set of vectors is orthogonal with respect to the standard inner product on \(M_{22}\) (see Example 6 of Section 6.1). a. \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{cc} 1 & \frac{2}{3} \\ \frac{2}{3} & -\frac{2}{3} \end{array}\right],\left[\begin{array}{cc} 0 & \frac{2}{3} \\ -\frac{2}{3} & \frac{1}{3} \end{array}\right],\left[\begin{array}{cc} 0 & \frac{1}{3} \\ \frac{2}{3} & \frac{2}{3} \end{array}\right] \) b. \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right],\left[\begin{array}{rr} 0 & 0 \\ 1 & -1 \end{array}\right] \) Equation Transcription: ????22 [], [], [], [] [], [], [], [] Text Transcription: M_22 [1 0 0 0], [1 2/3 2/3 -2/3], [0 2/3 -2/3 1/3], [0 1/3 2/3 2/3] [1 0 0 0], [0 1 0 0], [0 0 1 1], [0 0 1 -1]

In Exercises 5–6, show that the column vectors of ???? form an orthogonal basis for the column space of ???? with respect to the Euclidean inner product, and then find an orthonormal basis for that column space. \(A=\left[\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 0 & 5 \\ -1 & 2 & 0 \end{array}\right] \) Equation Transcription: A=[] Text Transcription: A=[1 2 0 0 0 5 -1 2 0]

In Exercises 5–6, show that the column vectors of ???? form an orthogonal basis for the column space of ???? with respect to the Euclidean inner product, and then find an orthonormal basis for that column space. \(A=\left[\begin{array}{ccc} \frac{1}{5} & -\frac{1}{2} & \frac{1}{3} \\ \frac{1}{5} & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{5} & 0 & -\frac{2}{3} \end{array}\right] \) Equation Transcription: A=[] Text Transcription: A=[1/5 -1/2 1/3 1/5 1/2 1/3 1/5 0 -2/3]

Verify that the vectors \(v_{1}=\left(-\frac{3}{5}, \frac{4}{5}, 0\right), v_{2}=\left(\frac{4}{5}, \frac{3}{5}, 0\right), v_{3}=(0,0,1)\) form an orthonormal basis for \(\mathrm{R}^{3}\) with respect to the Euclidean inner product, and then use Theorem 6.3.2(b) to express the vector \(u = (1, ?2, 2)\) as a linear combination of \(\mathrm{v}_{1}, \mathrm{v}_{2}\) , and \(\mathrm{v}_{3}\) . Equation Transcription: R3 u = (1, ?2, 2) v1 , v2 v3 Text Transcription: v_1=(-35, 45, 0), v_2=(45, 35, 0), v_3=(0, 0, 1) R^3 u = (1, ?2, 2) v_1 , v_2 v_3

Use Theorem 6.3.2(b) to express the vector \(u = (3, ?7, 4)\) as a linear combination of the vectors \(\mathrm{v}_{1}, \mathrm{v}_{2}\) , and \(\mathrm{v}_{3}\) in Exercise 7. Equation Transcription: u = (3, ?7, 4) v1 , v2 v3 Text Transcription: u = (3, ?7, 4) v_1 , v_2 v_3

Verify that the vectors \(\mathrm{v}_{1}=(2,-2,1), \quad \mathrm{v}_{2}=(2,1,-2), \quad \mathrm{v}_{3}=(1,2,2)\) form an orthogonal basis for \(\mathrm{R}^{3}\) with respect to the Euclidean inner product, and then use Theorem 6.3.2(a) to express the vector \(\mathrm{u}=(-1,0,2)\) as a linear combination of \(\mathrm{v}_{1}, \mathrm{v}_{2}\) , and \(\mathrm{v}_{3}\) . Equation Transcription: v1 = (2, ?2, 1), v2 = (2, 1, ?2), v3 = (1, 2, 2) R3 u = (?1, 0, 2) v1 , v2 v3 Text Transcription: v_1 = (2, ?2, 1), v_2 = (2, 1, ?2), v_3 = (1, 2, 2) R^3 u = (?1, 0, 2) v_1 , v_2 v_3

Verify that the vectors \(\mathrm{v}_{1}=(1,-1,2,-1), \quad \mathrm{v}_{2}=(-2,2,3,2)\), \(v_{3}=(1,2,0,-1), \quad v_{4}=(1,0,0,1)\) form an orthogonal basis for \(\mathrm{R}^{4}\) with respect to the Euclidean inner product, and then use Theorem 6.3.2(a) to express the vector \(\mathrm{u}=(1,1,1,1)\) as a linear combination of \(v_{1}, v_{2}, v_{3}\) , and \(v_{4}\) . Equation Transcription: v1 = (1, ?1, 2, ?1), v2 = (?2, 2, 3, 2) v3 = (1, 2, 0, ?1), v4 = (1, 0, 0, 1) R4 u = (1, 1, 1, 1) v1 , v2 , v3 v4 Text Transcription: v_1 = (1, ?1, 2, ?1), v_2 = (?2, 2, 3, 2) v_3 = (1, 2, 0, ?1), v_4 = (1, 0, 0, 1) R^4 u = (1, 1, 1, 1) v_1 , v_2 , v_3 v_4

In Exercises \(11-14\), find the coordinate vector \((u) S\) for the vector u and the basis \(S\) that were given in the stated exercise. Exercise 7 Equation Transcription: Text Transcription: 11–14 (u)S S

In Exercises \(11-14\), find the coordinate vector \((u) S\) for the vector u and the basis \(S\) that were given in the stated exercise. Exercise 8 Equation Transcription: Text Transcription: 11–14 (u)S S

In Exercises \(11-14\), find the coordinate vector \((u) S\) for the vector u and the basis \(S\) that were given in the stated exercise. Exercise 9 Equation Transcription: Text Transcription: 11–14 (u)S S

In Exercises \(11-14\), find the coordinate vector \((u) S\) for the vector u and the basis \(S\) that were given in the stated exercise. Exercise 10 Equation Transcription: Text Transcription: 11–14 (u)S S

In Exercises 15–18, let \(\mathrm{R}^{2}\) have the Euclidean inner product. a. Find the orthogonal projection of u onto the line spanned by the vector v. b. Find the component of u orthogonal to the line spanned by the vector v, and confirm that this component is orthogonal to the line \(\mathrm{u}=(-1,6) ; \mathrm{v}=\left(\frac{3}{5}, \frac{4}{5}\right)\) Equation Transcription: R2 u = (?1, 6); v = ( , ) Text Transcription: R^2 u = (?1, 6); v = ( 3/5 , 4/5 )

In Exercises 15–18, let \(\mathrm{R}^{2}\) have the Euclidean inner product. a. Find the orthogonal projection of u onto the line spanned by the vector v. b. Find the component of u orthogonal to the line spanned by the vector v, and confirm that this component is orthogonal to the line \(\mathrm{u}=(2,3) ; \mathrm{v}=\left(\frac{5}{13}, \frac{12}{13}\right)\) Equation Transcription: R2 u = (2, 3); v = ( , ) Text Transcription: R^2 u = (2, 3); v = (5/13 , 12/13 )

In Exercises 15–18, let \(\mathrm{R}^{2}\) have the Euclidean inner product. a. Find the orthogonal projection of u onto the line spanned by the vector v. b. Find the component of u orthogonal to the line spanned by the vector v, and confirm that this component is orthogonal to the line \(u=(2,3) ; v=(1,1)\) Equation Transcription: R2 u = (2, 3); v = (1, 1) Text Transcription: R^2 u = (2, 3); v = (1, 1)

In Exercises 15–18, let \(\mathrm{R}^{2}\) have the Euclidean inner product. a. Find the orthogonal projection of u onto the line spanned by the vector v. b. Find the component of u orthogonal to the line spanned by the vector v, and confirm that this component is orthogonal to the line \(u=(3,-1) ; v=(3,4)\) Equation Transcription: R2 u = (3, ?1); v = (3, 4) Text Transcription: R^2 u = (3, ?1); v = (3, 4)

In Exercises 19–22, let \(\mathrm{R}^{3}\). have the Euclidean inner product. a. Find the orthogonal projection of u onto the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) . b. Find the component of u orthogonal to the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) , and confirm that this component is orthogonal to the plane. \(\mathrm{u}=(4,2,1) ; \mathrm{v}_{1}=\left(\frac{1}{3}, \frac{2}{3},-\frac{2}{3}\right), \mathrm{v}_{2}=\left(\frac{2}{3}, \frac{1}{3}, \frac{2}{3}\right)\) Equation Transcription: R3 v1 v2 u = (4, 2, 1); v1 = ( , , - ), v2 = ( , , ) Text Transcription: R^3 v_1 v_2 u = (4, 2, 1); v_1 = ( 1/3 , 2/3 , - 2/3 ), v_2 = ( 2/3 , 1/3 , 2/3 )

In Exercises 19–22, let \(\mathrm{R}^{3}\). have the Euclidean inner product. a. Find the orthogonal projection of u onto the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) . b. Find the component of u orthogonal to the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) , and confirm that this component is orthogonal to the plane. \(u=(3,-1,2) ; v_{1}=\left(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}}\right), v_{2}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\) Equation Transcription: R3 v1 v2 Text Transcription: R^3 v_1 v_2 u=(3, -1, 2); v_1=(1/sqrt 6,1/sqrt 6,-2/sqrt 6), v_2=(1/sqrt 3,1/sqrt 3,1/sqrt 3)

In Exercises 19–22, let \(\mathrm{R}^{3}\). have the Euclidean inner product. a. Find the orthogonal projection of u onto the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) . b. Find the component of u orthogonal to the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) , and confirm that this component is orthogonal to the plane. \(u=(1,0,3) ; v_{1}=(1,-2,1), v_{2}=(2,1,0)\) Equation Transcription: R3 v1 v2 u = (1, 0, 3); v1 = (1, ?2, 1), v2 = (2, 1, 0) Text Transcription: R^3 v_1 v_2 u = (1, 0, 3); v_1 = (1, ?2, 1), v_2 = (2, 1, 0)

In Exercises 19–22, let \(\mathrm{R}^{3}\). have the Euclidean inner product. a. Find the orthogonal projection of u onto the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) . b. Find the component of u orthogonal to the plane spanned by the vectors \(v_{1}\) and \(v_{2}\) , and confirm that this component is orthogonal to the plane. \(u=(1,0,2) ; v_{1}=(3,1,2), v_{2}=(-1,1,1)\) Equation Transcription: R3 v1 v2 u = (1, 0, 2); v1 = (3, 1, 2), v2 = (?1, 1, 1) Text Transcription: R^3 v_1 v_2 u = (1, 0, 2); v_1 = (3, 1, 2), v_2 = (?1, 1, 1)

In Exercises 23–24, the vectors \(v_{1}\) and \(v_{2}\) are orthogonal with respect to the Euclidean inner product on \(\mathrm{R}^{4}\). Find the orthogonal projection of \(\mathrm{b}=(1,2,0,-2)\) on the subspace ???? spanned by these vectors. \(\mathrm{v}_{1}=(1,1,1,1), \mathrm{v}_{2}=(1,1,-1,-1)\) Equation Transcription: v1 v2 R4 b = (1, 2, 0, ?2) v1 = (1, 1, 1, 1), v2 = (1, 1, ?1, ?1) Text Transcription: v_1 v_2 R^4 b = (1, 2, 0, ?2) v_1 = (1, 1, 1, 1), v_2 = (1, 1, ?1, ?1)

In Exercises 23–24, the vectors \(v_{1}\) and \(v_{2}\) are orthogonal with respect to the Euclidean inner product on \(\mathrm{R}^{4}\). Find the orthogonal projection of \(\mathrm{b}=(1,2,0,-2)\) on the subspace ???? spanned by these vectors. \(v_{1}=(0,1,-4,-1), v_{2}=(3,5,1,1)\) Equation Transcription: v1 v2 R4 b = (1, 2, 0, ?2) v1 = (0, 1, ?4, ?1), v2 = (3, 5, 1, 1) Text Transcription: v_1 v_2 R^4 b = (1, 2, 0, ?2) v_1 = (0, 1, ?4, ?1), v_2 = (3, 5, 1, 1)

In Exercises 25–26, the vectors \(v_{1}, v_{2}\) , and \(v_{3}\) are orthonormal with respect to the Euclidean inner product on \(\mathrm{R}^{4}\) . Find the orthogonal projection of \(b=(1,2,0,-1)\) onto the subspace ???? spanned by these vectors. \(v_{1}=\left(0, \frac{1}{\sqrt{18}},-\frac{4}{\sqrt{18}},-\frac{1}{\sqrt{18}}\right), v_{2}=\left(\frac{1}{2}, \frac{5}{6}, \frac{1}{6}, \frac{1}{6}\right)\) \(v_{3}=\left(\frac{1}{\sqrt{18}}, 0, \frac{1}{\sqrt{18}},-\frac{4}{\sqrt{18}}\right)\) Equation Transcription: v1 , v2 v3 R4 Text Transcription: v_1 , v_2 v_3 R^4 b = (1, 2, 0, ?1) v_1=(0, 1/sqrt 18,-4/sqrt 18,-1/sqrt 18), v_2=(1/2,5/6,1/6,1/6), v_3=(1/sqrt 18,0,1/sqrt 18,-4/sqrt 18)

In Exercises 25–26, the vectors \(v_{1}, v_{2}\) , and \(v_{3}\) are orthonormal with respect to the Euclidean inner product on \(\mathrm{R}^{4}\) . Find the orthogonal projection of \(b=(1,2,0,-1)\) onto the subspace ???? spanned by these vectors. \(v_{1}=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right), v_{2}=\left(\frac{1}{2}, \frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right), v_{3}=\left(\frac{1}{2},-\frac{1}{2}, \frac{1}{2},-\frac{1}{2}\right)\) Equation Transcription: v1 , v2 v3 R4 Text Transcription: v_1 , v_2 v_3 R^4 b = (1, 2, 0, ?1) v_1=(1/2,1/2,1/2,1/2), v_2=(1/2,1/2,-1/2,-1/2), v_3=(1/2,-1/2,1/2,-1/2)

In Exercises 27–28, let \(R^{2}\) have the Euclidean inner product and use the Gram–Schmidt process to transform the basis {\(\mathrm{u}_{1}, \mathrm{u}_{2}\)} into an orthonormal basis. Draw both sets of basis vectors in the xy-plane. \(\mathrm{u}_{1}=(1,-3), \mathrm{u}_{2}=(2,2)\) Equation Transcription: R2 u1 , u2 u1 = (1, ?3), u2 = (2, 2) Text Transcription: R^2 u_1 , u_2 u_1 = (1, ?3), u_2 = (2, 2)

In Exercises 27–28, let \(R^{2}\) have the Euclidean inner product and use the Gram–Schmidt process to transform the basis {\(\mathrm{u}_{1}, \mathrm{u}_{2}\)} into an orthonormal basis. Draw both sets of basis vectors in the xy-plane. \(\mathrm{u}_{1}=(1,0), \quad \mathrm{u}_{2}=(3,-5)\) Equation Transcription: R2 u1 , u2 u1 = (1, 0), u2 = (3, ?5) Text Transcription: R^2 u_1 , u_2 u_1 = (1, 0), u_2 = (3, ?5)

In Exercises 29–30, let \(R^{3}\) have the Euclidean inner product and use the Gram–Schmidt process to transform the basis {\(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}\)} into an orthonormal basis. \(u_{1}=(1,1,1), u_{2}=(-1,1,0), u_{3}=(1,2,1)\) Equation Transcription: R3 u1 , u2 , u3 u1 = (1, 1, 1), u2 = (?1, 1, 0), u3 = (1, 2, 1) Text Transcription: R^3 u_1 , u_2 , u_3 u_1 = (1, 1, 1), u_2 = (?1, 1, 0), u_3 = (1, 2, 1)

In Exercises 29–30, let \(R^{3}\) have the Euclidean inner product and use the Gram–Schmidt process to transform the basis {\(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}\)} into an orthonormal basis. \(\mathrm{u}_{1}=(1,0,0), \mathrm{u}_{2}=(3,7,-2), \mathrm{u}_{3}=(0,4,1)\) Equation Transcription: R3 u1 , u2 , u3 u1 = (1, 0, 0), u2 = (3, 7, ?2), u3 = (0, 4, 1) Text Transcription: R^3 u_1 , u_2 , u_3 u_1 = (1, 0, 0), u_2 = (3, 7, ?2), u_3 = (0, 4, 1)

Let \(R^{4}\) have the Euclidean inner product. Use the Gram– Schmidt process to transform the basis {\(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}, \mathrm{u}_{4}\) } into an orthonormal basis. \(\mathrm{u}_{1}=(0,2,1,0), \quad \mathrm{u}_{2}=(1,-1,0,0)\), \(\mathrm{u}_{3}=(1,2,0,-1), \quad \mathrm{u}_{4}=(1,0,0,1)\) Equation Transcription: R4 u1 , u2 , u3 , u4 u1 = (0, 2, 1, 0), u2 = (1, ?1, 0, 0) u3 = (1, 2, 0, ?1), u4 = (1, 0, 0, 1) Text Transcription: R^4 u_1 , u_2 , u_3 , u_4 u_1 = (0, 2, 1, 0), u_2 = (1, ?1, 0, 0) u_3 = (1, 2, 0, ?1), u_4 = (1, 0, 0, 1)

Let \(R^{3}\) have the Euclidean inner product. Find an orthonormal basis for the subspace spanned by \((0,1,2),(-1,0,1),(-1,1,3)\). Equation Transcription: R3 (0, 1, 2), (?1, 0, 1), (?1, 1, 3) Text Transcription: R^3 (0, 1, 2), (?1, 0, 1), (?1, 1, 3)

Let b and ???? be as in Exercise 23. Find vectors \(w^{1}\) in ???? and \(w^{2}\) in \(W^{\perp}\) such that \(\mathrm{b}=\mathrm{w}_{1}+\mathrm{W}_{2}\) . Equation Transcription: w1 w2 b = w1 + w2 Text Transcription: w_1 w_2 ????^prep b = w_1 + w_2

Let b and ???? be as in Exercise 25. Find vectors \(w^{1}\) in ???? and \(w^{2}\) in \(W^{\perp}\) such that \(\mathrm{b}=\mathrm{w}_{1}+\mathrm{W}_{2}\) . Equation Transcription: w1 w2 b = w1 + w2 Text Transcription: w_1 w_2 ????^prep b = w_1 + w_2

Let \(R^{3}\) have the Euclidean inner product. The subspace of \(R^{3}\) spanned by the vectors \(\mathrm{u}_{1}=(1,1,1)\) and \(\mathrm{u}_{2}=(2,0,-1)\) is a plane passing through the origin. Express \(w=(1,2,3)\) in the form \(\mathrm{w}=\mathrm{w}_{1}+\mathrm{w}_{2}\) , where \(w^{1}\) lies in the plane and \(w^{2}\) is perpendicular to the plane. Equation Transcription: ???? 3 u1 = (1, 1, 1) u2 = (2, 0, ?1) w = w1 + w2 w1 w2 Text Transcription: u_2 = (2, 0, ?1) w = (1, 2, 3) w = w_1 + w_2 w_1 w_2

Let \(R^{4}\) have the Euclidean inner product. Express the vector \(w = (?1, 2, 6, 0)\) in the form \(w=w_{1}+w_{2}\), where \(w_{1}\)is in the space \(W\) that is spanned by \(u_{1}=(-1,0,1,2)\) and \(u_{2}=(0,1,0,1)\) and \(w_{2}\) is orthogonal to \(W\). Equation Transcription: Text Transcription: R^4 w = (?1, 2, 6, 0) w = w_1+ w_2 W u_1= (?1, 0, 1, 2) u_2= (0, 1, 0, 1)

Let \(R^{3}\) have the inner product \(\langle u, v\rangle=u_{1} v_{1}+2 u_{2} v_{2}+3 u_{3} v_{3}\) Use the Gram–Schmidt process to transform \(\mathrm{u}_{1}=(1,1,1), \mathrm{u}_{2}=(1,1,0), \mathrm{u}_{3}=(1,0,0)\) into an orthonormal basis. Equation Transcription: ?? Text Transcription: R^3 ?u,v?=u_1v_1 +2u_2v_2 + 3u_3v_3 u_1=(1, 1, 1) u_2= (1, 1, 0) u_3= (1, 0, 0)

Verify that the set of vectors \({(1, 0), (0, 1)}\) is orthogonal with respect to the inner product \(\langle u, v\rangle=4 u_{1} v_{1}+2 u_{2} v_{2}\) on \(R^{2}\) ; then convert it to an orthonormal set by normalizing the vectors. Equation Transcription: ?? Text Transcription: {(1, 0), (0, 1)} ?u,v?=4u_1v_1 +2u_2v_2 R^2

Find vectors \(x\) and \(y\) in \(R^{2}\) that are orthonormal with respect to the inner product \(\langle u, v\rangle=3 u_{1} v_{1}+2 u_{2} v_{2}\) but are not orthonormal with respect to the Euclidean inner product. Equation Transcription: ?? Text Transcription: ?u,v?=3u_1v_1 +2u_2v_2 R^2 x y

In Example 6 of Section 3.3 we found the orthogonal projection of the vector \(x = (1, 5)\) onto the line through the origin making an angle of \(\pi / 6\) radians with the positive \(x\)-axis. Solve that same problem using Theorem 6.3.4. Equation Transcription: Text Transcription: pi/6 x= (1, 5) x

This exercise illustrates that the orthogonal projection resulting from Formula (12) in Theorem 6.3.4 does not depend on which orthogonal basis vectors are used. a. Let \(R^{3}\) have the Euclidean inner product, and let ???? be the subspace of \(R^{3}\) spanned by the orthogonal vectors \(v_{1}=(1,0,1)\) and \(v_{2}=(0,1,0)\) Show that the orthogonal vectors \(v_{1}^{\prime}=(1,1,1)\) and \(v_{2}^{\prime}=(1,-2,1)\) span the same subspace \(W\). b. Let \(u = (?3, 1, 7)\) and show that the same vector \(\text { proj }_{w} u\) results regardless of which of the bases in part (a) is used for its computation. Equation Transcription: Text Transcription: R^3 v_1= (1, 0, 1) v_2= (0, 1, 0) v ? _1= (1, 1, 1) v ? _2= (1, ?2, 1) W u = (?3, 1, 7) proj_W u

(Calculus required) Use Theorem 6.3.2(a) to express the following polynomials as linear combinations of the first three Legendre polynomials (see the Remark following Example 9). \(a. 1+x+4 x^{2}\) \(b. 2-7 x^{2}\) \(c. 4+3 x\) Equation Transcription: Text Transcription: 1+x+4x^2 2-7x^2 4+3x

(Calculus required) Let \(P_{2}\) have the inner product \(\langle\mathbf{p}, \mathbf{q}\rangle=\int_{0}^{1} p(x) q(x)\ d x\) Apply the Gram–Schmidt process to transform the standard basis \(S=\left\{1, x, x^{2}\right\}\) into an orthonormal basis. Equation Transcription: ?? Text Transcription: ?p,q?=integral_0^1 p(x)q(x) dx ???? = {1,x,x^2} P_2

Find an orthogonal basis for the column space of the matrix \(A=\left[\begin{array}{rrr} 6 & 1 & -5 \\ 2 & 1 & 1 \\ -2 & -2 & 5 \\ 6 & 8 & -7 \end{array}\right]\) Equation Transcription: [ ] Text Transcription: A=[6 & 1 & -5 \\ 2 & 1 & 1 \\ -2 & -2 & 5 \\ 6 & 8 & -7]

In Exercises 45–48, we obtained the column vectors of \(Q\) by applying the Gram–Schmidt process to the column vectors of \(A\). Find a \(QR\)-decomposition of the matrix \(A\). \(A=\left[\begin{array}{rr} 1 & -1 \\ 2 & 3 \end{array}\right], \quad Q=\left[\begin{array}{ll} \frac{1}{\sqrt{5}} & -\frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{array}\right]\) Equation Transcription: Text Transcription: Q A QR A=[2 & 3\\1 & -1] Q=[2/square root 5 & 1/square root 5\\1/square root 5 & -2/square root 5]

In Exercises 45–48, we obtained the column vectors of \(Q\) by applying the Gram–Schmidt process to the column vectors of \(A\). Find a \(QR\)-decomposition of the matrix \(A\). \(A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 1 \\ 1 & 4 \end{array}\right], Q=\left[\begin{array}{cc} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}\right]\) Equation Transcription: Text Transcription: Q A QR A=[1 & 2 \\ 0 & 1 \\ 1 & 4], Q=[1/square root 2 & 1/square root 3\\ 0 & 1/square root 3 \\ 1/square root 2 & \1/square root 3]

In Exercises 45–48, we obtained the column vectors of \(Q\) by applying the Gram–Schmidt process to the column vectors of \(A\). Find a \(QR\)-decomposition of the matrix \(A\). \(A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 1 & 2 & 0 \end{array}\right], Q=\left[\begin{array}{ccc} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} \\ 0 & \frac{1}{\sqrt{3}} & \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{6}} \end{array}\right]\) Equation Transcription: = [], = [] Text Transcription: Q A QR A=[1 & 0 & 2 \\ 0 & 1 & 1 \\ 1 & 2 & 0] Q=[1/square root 2 & -1/square root 3 & 1/square root 6\\ 0 & 1/square root 3 & 1/square root 6 \\1/square root 2 & 1/square root 3 & -1/square root 6]

In Exercises 45–48, we obtained the column vectors of \(Q\) by applying the Gram–Schmidt process to the column vectors of \(A\). Find a \(QR\)-decomposition of the matrix \(A\). \(A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 1 & 1 & 1 \\ 0 & 3 & 1 \end{array}\right], Q=\left[\begin{array}{ccc} \frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{2 \sqrt{19}} & -\frac{3}{\sqrt{19}} \\ \frac{1}{\sqrt{2}} & -\frac{\sqrt{2}}{2 \sqrt{19}} & \frac{3}{\sqrt{19}} \\ 0 & \frac{3 \sqrt{2}}{\sqrt{19}} & \frac{1}{\sqrt{19}} \end{array}\right]\) Equation Transcription: = [], = [] Text Transcription: Q A QR A=[1 & 2 & 1 \\ 1 & 1 & 1 \\ 0 & 3 & 1] Q=[1/square root 2 & square root 2/2 square root 19 & -3/square root 19\\ 1/square root 2 & -square root 2/2 square root 19 & 3/square root 19 \\ 0 & 3 square root 2/square root 19 &1/square root 19]

Find a \(QR\)-decomposition of the matrix \(A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ -1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: QR A=[1 & 0 & 1 \\ -1 & 1 & 1 \\ 1 & 0 & 1 \\ -1 & 1 & 1]

In the Remark following Example 8 we discussed two alternative ways to perform the calculations in the Gram–Schmidt process: normalizing each orthogonal basis vector as soon as it is calculated and scaling the orthogonal basis vectors at each step to eliminate fractions. Try these methods in Example 8.

Prove part (a) of Theorem 6.3.6.

In Step 3 of the proof of Theorem 6.3.5, it was stated that “the linear independence of \(\left\{u_{1}, u_{2}, \cdots,, u_{n}\right\}\) ensures that \(v_{3} \neq 0\) .” Prove this statement. Equation Transcription: Text Transcription: {u_1 , u_2 , . . . , u_n } V_3 not equal 0

Prove that the diagonal entries of \(R\) in Formula (16) are nonzero. Equation Transcription: Text Transcription: R

Show that matrix \(Q\) given in Example 10 satisfies the equation \(Q Q^{T}=l_{3}\), and prove that every \(m \times n\) matrix \(Q\) with orthonormal column vectors has the property \(Q Q^{T}=l_{m}\). Equation Transcription: Text Transcription: Q ????????^T= ????_3 m times n ????????^T= ????_m

a. Prove that if ???? is a subspace of a finite-dimensional vector space \(V\), then the mapping \(T: V \rightarrow W\) that is defined by \(T(v)=\operatorname{proj}_{w} v\) is a linear transformation. b. What are the range and kernel of the transformation in part (a)? Equation Transcription: Text Transcription: V ???? ? ????->???? ????(v) = proj_w v

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. a. Every linearly independent set of vectors in an inner product space is orthogonal.

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. b. Every orthogonal set of vectors in an inner product space is linearly independent.

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. c. Every nontrivial subspace of \(R^{3}\) has an orthonormal basis with respect to the Euclidean inner product. Equation Transcription: Text Transcription: R^3

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. d. Every nonzero finite-dimensional inner product space has an orthonormal basis.

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. e.\({proj}_{w} x\) is orthogonal to every vector of \(W\). Equation Transcription: Text Transcription: proj_w x W

In parts (a)–(f ) determine whether the statement is true or false, and justify your answer. f. If \(A\) is an \(n \times n\) matrix with a nonzero determinant, then \(A\) has a \(QR\)-decomposition. Equation Transcription: Text Transcription: A n times n QR

a. Use the Gram–Schmidt process to find an orthonormal basis relative to the Euclidean inner product for the column space of \(A=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 2 & -1 & 1 & 1 \end{array}\right]\) b. Use the method of Example 9 to find a \(????????\)-decomposition of \(A\). Equation Transcription: [ ] Text Transcription: A=[1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 2 & -1 & 1 & 1] QR A

Let \(p_{4}\) have the evaluation inner product at the points \(?2, ?1, 0, 1, 2\). Find an orthogonal basis for \(p_{4}\) relative to this inner product by applying the Gram–Schmidt process to the vectors \(p_{0}=1, p_{1}=x, p_{2}=x^{2}, p_{3}=x^{3}, p_{4}=x^{4}\) Equation Transcription: Text Transcription: ?2, ?1, 0, 1, 2 p_0= 1, p_1= x, p_2 = x^2, p_3= x^3, p_4= x^ 4 p_4