Only Part D neededNeeded to be solved Part A Correctly in 15 minutes and get the thumbs up please show neat and clean work for it 1. a)b)c)DLiii)ii)iii)iv)Use the maple output at the end of this question to give answers to the followingquestions:1-i)ii)3] Let b (b1,b2, b3, b₁) € R4 and(-col(A).ii)Suppose that A € Mkn (R). Define the following terms:the column space of the matrix A, col(A) CR.the nullspace of the matrix A, null (A) CR"{null (A).col(A)?PRAUse the maple outputquestions:7-A =s] Find the basis and the dimension of the column space of the matrix A,3] Find the basis and the dimension of the nullspace of the matrix A,:] What are the conditions on the components of the vector b such that bes] Find the coordinate vector of the quadruple c = (1, 3, -2, 3)¹ € col (4)with respect to the basis you gave in response to part i) above.Let-11 33-3 -300-2-1 1 5of 3 x 3B =11 2--0--0and y=1 34.REY-] Find the projection of the vector y onto the subspace col (B).s] Let Q be the matrix in the maple output (4) at the end of this question.Explain why QQT = B(BTB) ¹BT.•iii).best fits the points (1, 1), (2,0),Explain why the line y = - +(3, 2), (4,5) in the least squares sense.Let x = (1, 22, 23)b= (b₁,b2, b) € C³ be fixed. LetC³ be arbitrary and let a = (a₁, 02, 03) € C³,2at the end of this question to give answers to the followingT(x) = deta₁ a2 a3b1 b2 b3x1 12 13/Find T(3e3).Use the symbol T(x) to explain what it means that the determinantmatrix is a linear map with respect to the third row.Use the result in the preceding part, or otherwise, to prove that T(x) =0, if x span {a, b}. Hint: You can use without proof the fact that the determinantof a 3 x 3 matrix with two identical rows is zero.INSW

Answered: Image /qna-images/answer/e9905663-0d94-475f-9267-970b2b2cc41b.jpg

d) Let x = (1, 2, 3) C3 be arbitrary and let a = (a1, 02, 03) € C³, b= (b₁,b2, b) C³ be fixed. Let a1 a2 a3 TH-4639 T(x)= det b₁ b₂ b3 22 23/ Find T(303). ii) Use the symbol T(x) to explain what it means that the determinant of 3 x 3 matrix is a linear map with respect to the third row. iii) Use the result in the preceding part, or otherwise, to prove that T(x) = 0, if x E span {a, b}. Hint: You can use without proof the fact that the determinant of a 3 x 3 matrix with two identical rows is zero.

d) Let x = (1, 2, 3) C3 be arbitrary and let a = (a1, 02, 03) € C³, b= (b₁,b2, b) C³ be fixed. Let a1 a2 a3 TH-4639 T(x)= det b₁ b₂ b3 22 23/ Find T(303). ii) Use the symbol T(x) to explain what it means that the determinant of 3 x 3 matrix is a linear map with respect to the third row. iii) Use the result in the preceding part, or otherwise, to prove that T(x) = 0, if x E span {a, b}. Hint: You can use without proof the fact that the determinant of a 3 x 3 matrix with two identical rows is zero.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 6AEXP

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