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 - 2x³, 24 = 1+ 2x - 3x² + 3x³, vs = -2-5x + 5x2 - 4x3. Which of the following statements are true? (i) The set Bw = {₁, 3,₁) is a basis for W. (ii) The set Cw = {₁,4} is a basis for W. (iii) The set By = (v₁, 3, 1, 2} is a basis for the space of polynomials of degree 3. (iv) The set Cv = {v1, vs, 1, 22} is a basis for the space of polynomials of degree 3. (v) a+bx+cx² + da 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. OA) () and (v) OB) (i), (i), and (iv) OC)(1). (i) and (vi) OD) (ii) and (lv) E) () (lv), and (v) 27
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 - 2x³, 24 = 1+ 2x - 3x² + 3x³, vs = -2-5x + 5x2 - 4x3. Which of the following statements are true? (i) The set Bw = {₁, 3,₁) is a basis for W. (ii) The set Cw = {₁,4} is a basis for W. (iii) The set By = (v₁, 3, 1, 2} is a basis for the space of polynomials of degree 3. (iv) The set Cv = {v1, vs, 1, 22} is a basis for the space of polynomials of degree 3. (v) a+bx+cx² + da 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. OA) () and (v) OB) (i), (i), and (iv) OC)(1). (i) and (vi) OD) (ii) and (lv) E) () (lv), and (v) 27
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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)
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