Let W be the subspace of R³ spanned by the vectors1-2Find the matrix A of the orthogonal projection onto W.A =and8-3-8 LetConsider the inner productf(x) = -5, g(x)=5x+2 and h(x) = 2x².(p,q9) = f* p(x)q(x) dæin the vector space P7 of polynomials of degree at most 7. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of P7 spanned by the functions f(x), g(x), and h(x).{100}.

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Answered: Let W be the subspace of R³ spanned by…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Find the projection of the vector v=[102]T onto the subspace S=span{[011],[011]}.
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector space?
Find a basis for R2 that includes the vector (2,2).
True or False, S is a subspace of R 3. x1 S = x2 x3 O True O False 2x1 + x3 = 0, x1 + x2 = x3 = 0 +32-33=0}
Using linear algebra, show that G={p1(x)=1−x+x^2, p2(x)=x, p3(x)=x+x^2} is a basis of P2, the vector space of polynomials of degree at most two with real coefficients.
Let P₂ be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 3x 6x²4, 2x11x² 6 and a. The dimension of the subspace H is b. Is {3x6x²4, 2x11x²-6, x - x² c. A basis for the subspace H is { 1} a basis for P₂? choose ✓ Be sure you can explain and justify your answer. ²1. }. Enter a polynomial or a comma separated list of polynomials (where you can enter xx in place of ²)
Let f(z) = -7, g(z) = -3z - 9 and h(z) = 8z". Consider the inner procuct (p, 9) in the vector space P(R) of polynomials. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace spanned by the functions f(2) 9(z), and h(z). {
Let W be the subspace of the vector space of polynomials of degree 3 spanned by 2₁ = 1 + 3x - 2x² + x³, v₂ = -3-9x + 6x²-3x³, v3 = x + x². x+x² - 2x³, 24 = 1+ 2x - 3x² + 3x³, vs = -2-5x + 5x2 - 4x3. Which of the following statements are true? (i) The set Bw = {₁, 23,1₁) is a basis for W. (ii) The set Cw = {₁, 4) is a basis for W. (iii) The set By = {1, 23, 1, 2} is a basis for the space of polynomials of degree 3. (iv) The set Cy (v1,vs, 1,22} is a basis for the space of polynomials of (v) a +bx+cx² + da3 is in W, if 5a - b + d = 0 and 7a+2b+c=0 (vi) 1-x+x²-23 EW and -1-4x+x² + x³ € W. degree 3. A) () and (v) B) (i), (iii) and (iv) C) (i), (ii), and (vi) OD) (ii) and (iv) E) (), (iv) and (v)
Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, andS= {1 + 2x, 1 + 3x + 5x^2 , 4x + 5x^2 } .Is Span(S)= P2?

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