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a. Fill in the blank in the following statement: "If $A$ is an $m \times n$ matrix, then the columns of $A$ are linearly independent if and only if $A$ has ______ pivot columns.”b. Explain why the statement in (a) is true.

$n$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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in this video, we're gonna be solving problem number 30 from section 1.7 of the book, which is based on a linear independence. So, um, I'm not gonna write the problem out here, but it has two parts. The first part is a fill in the blank which asked you to fill in the blank if ah, for the following statement. If a is an m by N Matrix, then the columns of a r linearly independent if and only if they has dash pivot Collins so is and by and it is linearly independent. Yeah, okay. As blank pivot columns, then B is the explanation of why it's always true. So a the phone The blank is, um is And if a is an n by n matrix its columns, they're only linearly independent. If has n pivot columns this because if he has an pickles will say we have a name by and cellos pick up two by three matrix for number 10. So here m is too. I just pick a four by three. Um So m is for, in this case, an n equals three. So here, if you have one euro on star So here, Um, here we have a leading Are you here? We have a pivot position in every column. So, um so this. Make sure that there is no free variable. Final answer. So if let's say if let's say there were only two pivots and this was zero, that means this column would have been a free variable making the entire column the columns of this matrix b linearly dependent Because, um, these two variables will be dependent on the value of the free variable. So for that snobby, the case, like an M by N. Matrix, has tohave end pivot columns so that there are no free variables possible so that the only solution, um to the matrix equation A X equals zero as a trivial solution, which is, Ah, zero vector. Is there a column vector depending on

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