5. Consider the vector space P2(R) together with inner product =P(-1),(-1) + p(0,0) + p(1)g(1). (5) (a) Use the Gram-Schmidt procedure to find an orthogonal basis for W = Span{1, x + 32²}. [5] (b) Find the orthogonal projection of 2 + 4x onto W.

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mdasfaaruddinshareefLv10

13 Jun 2023

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Lelia LubowitzLv2

9 Aug 2018

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4. Let S= 1 1 , and consider the subspace W = Span(S) of R. (a) Show that S is an orthogonal basis for W. (b) Let x = 2. Find projw(x). 11 (c) Let x = 12 be an arbitrary vector in R3. Find projw(x) and then find a matrix A such that projw(x) = Ax.

and .

Consider the inner product ⟨𝐴,𝐵⟩=trace(𝐴𝑇𝐵) ⟨ A , B ⟩ = t r a c e ( A T B ) in the vector space ℝ2×2 R 2 × 2 of 2×2 2 × 2 matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of ℝ2×2 R 2 × 2 spanned by the matrices 𝑀1 M 1 and 𝑀2 M 2 .

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8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

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5. Consider the vector space P2(R) together with inner product =P(-1),(-1) + p(0,0) + p(1)g(1). (5) (a) Use the Gram-Schmidt procedure to find an orthogonal basis for W = Span{1, x + 32²}. [5] (b) Find the orthogonal projection of 2 + 4x onto W.

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