Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 4 - Vector Spaces - 4.5 Exercises - Page 231: 18

Answer

We have one free variable $Column\,{3}$. $\mathbf{dimension\,of\,Nul\,A\,is\,1}$ We have two pivotal columns $Column\,{1},Column\,{2}\,\mathbf{dimension\,of\,Col\,A\,is \,2}$

Work Step by Step

We are required to determine the dimensions of Nul A and Col A for the given matrix. $\mathbf{A}=\begin{bmatrix}{ 1 } & { 4 } & { - 1 } \\ { 0 } & { 7 } & { 0 } \\ { 0 } & { 0 }&0\end{bmatrix} $ The given matrix is already in echelon form. The dimension of Nul A is the number of free variables in the equation$ [AX=0]$ We have one free variables $Column\,{3}$. $\mathbf{dimension\,of\,Nul\,A\,is\,1}$ The dimension of Col A is the number of pivot columns in A. we have two pivotal columns $Column\,{1},Column\,{2}\,\mathbf{dimension\,of\,Col\,A\,is \,2}$
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