Complete parts a and b below for the matrix A.- 496328 - 35 - 21214928 - 42 - 49144235 - 35A=- 354942 - 3542 - 14 - 5621 - 35 - 5674928 - 56- 425635 - 28 - 28 - 63 -21a. Construct matrices C and N whose columns are bases for Col A and Nul A, respectively, and construct a matrix R whose rows form a basis for Row A.c=0C =N=OR=b. Construct a matrix M whose columns form a basis for Nul A', form the matrices S= RT N and T= C M, and explain why S and T should be square. Verify that both S and T are invertible.M=The matrix S= RT N isx because the columns of RT and N are in RU and dim Row A + dim Nul A =.[RT N]#The matrix T=[C M ]is xO because the columns of C and M are in R and dim Col A + dim Nul Aby the Rank Theorem, since Col A = Row AT.The columns of the matrix S are linearlyThe columns of S form a basis of RThus, by the Invertible Matrix Theorem, S is invertible.The columns of the matrix T are linearlyThe columns of T form a basis of RU. Thus, by the Invertible Matrix Theorem, T is invertible.

find the row reduced echelon form of A. To get the row space, consider the non zero rows of RREF(A).…

Complete parts a and b below for the matrix A. -49 63 28 - 35 - 21 21 49 28 - 42 - 49 14 42 35 - 35 A= - 35 49 42 - 35 42 - 14 - 56 21 - 35 - 56 7 49 28 - 56 -42 56 35 - 28 - 28- 63 -21 a. Construct matrices C and N whose columns are bases for Col A and NulA, respectively, and construct a matrix R whose rows form a basis for Row A. c-O N=O R= b. Construct a matrix M whose columns form a basis for Nul A", form the matrices S=[RT N]and T=[c m], and explain why S and T should be square. Verify that both S and T are invertible. M=[ The matrix S= [RT N]is -O because the columns of R"and N are in Rand dim Row A+ dim Nul A=D. The matrix T=[CM]isI×O because the columns of C and M are in R and dim Col A + dim Nul AT =O by the Rank Theorem, since Col A = Row A".

Complete parts a and b below for the matrix A. -49 63 28 - 35 - 21 21 49 28 - 42 - 49 14 42 35 - 35 A= - 35 49 42 - 35 42 - 14 - 56 21 - 35 - 56 7 49 28 - 56 -42 56 35 - 28 - 28- 63 -21 a. Construct matrices C and N whose columns are bases for Col A and NulA, respectively, and construct a matrix R whose rows form a basis for Row A. c-O N=O R= b. Construct a matrix M whose columns form a basis for Nul A", form the matrices S=[RT N]and T=[c m], and explain why S and T should be square. Verify that both S and T are invertible. M=[ The matrix S= [RT N]is -O because the columns of R"and N are in Rand dim Row A+ dim Nul A=D. The matrix T=[CM]isI×O because the columns of C and M are in R and dim Col A + dim Nul AT =O by the Rank Theorem, since Col A = Row A".

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 23EQ

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