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For problems 45-47, a subspace
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- Example S = {q1 = 2 + x + 4x², q2 = 1 - x + 3x², q3 = 3 + 2x + 5x²} a. Show that S is the basis for the polynomial space P2 b. Express q = 3-2x + 9x² as a linear combination of vectors in S c. Find the coordinate vector q relative to the base S or (q)sarrow_forwardBased on other problems I believe the dimension is 3, but I'm not sure how to find a basis given this subspace.arrow_forwardIn the vector space P of polynomials of degree at most two,the 2 second column of the A= (a₁ = ¹; a₂ = x; a ₂ = x²) to 1; 2 3 B=(e₁=2+x;e₂=1-x+6x²; e₁=1-x-6x² is: z= A. 1 - 1 -6 B. 1 ( ⁰9 08 1 C. 0 1 0 1 (:) 6 matrix of transition from the basis the basis O D.arrow_forward
- Find a basis of the subspace of R³ defined by the equation 2x₁ + 3x2 + x3 = 0.arrow_forwardConsider the following matrix and vector: 1 1 1 -4 0 -3 A = -2 and x= x3 Given that Ax=0 and x3=-1, find x2. O-1 O1 2 O-2arrow_forwardGive a formula for (ABx)T, where x is a vector and A and B are matrices of appropriate size. Choose the correct answer below. A. (ABx)T = XTATBT, because (ABx)T = xT (AB)T = XTATBT B. (ABx)T = ATBTX", because (ABx)T = (AB)TxT = ATBTXT C. (ABx)T = XTBTAT, because (ABx)T = x (AB)T = XTBTAT O D. (ABx)T = BTATXT, because (ABx)T = (AB)TxT = BTATXTarrow_forward
- Let A= (a,,a2,a3} and B= (b,.b2.b3} be bases for a vector space V, and suppose b, = 6a, - 3a3, b2 = - a, + 5a2, b3 = a, +a2 +2a3. a. Find the change-of-coordinates matrix from B to A. b. Find (x], for x = b, - 5b2 + 5b3. a. P = A-B b. [x]A = (Simplify your answers.)arrow_forwardThe integers 1 through 36 (inclusive) are used once each as the coefficients for six vectors in P 5 . What is the smallest dimensional subspace that can be spanned by the six vectors? Wholly justify your answer.arrow_forwardFind a basis B for the span of the given vectors. B= [01 1 ], [ 5 1 −1 0], [4 1 9 1] 0 1-4 1arrow_forward
- Let 8= {b,,b2} and C= {c1.c2} be bases for a vector space V, and suppose b, = -9c, +8c2 and b, = - 3c, + 7c2. a. Find the change-of-coordinates matrix from B to C. b. Find (x]c for x= 3b, - 6b2. Use part (a).arrow_forwardDetermine the dimensions of the following subspaces of R4 All vectors of the form (a, b, c, d), where d = a + b y c = a – b. All vectors of the form (a, b, c, d), where a = b = c = d. Note: In the image the problem is described more clearly, do not skip any step and solve the two parts a and b.arrow_forwardFind a set of vectors that spans the subspace WCV of solutions to the following linear systems. 1. 2 *1 + 2x₂ − x3 + £₁ = 0 -3x₁ — T3 + 2x4 = 0 |T₁ + 2x3 + x₁ = 0 Aarrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage