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In Exercises $5-8,$ determine if the columns of the matrix form a linearly independent set. Justify each answer.$\left[\begin{array}{rrrr}{1} & {4} & {-3} & {0} \\ {-2} & {-7} & {5} & {1} \\ {-4} & {-5} & {7} & {5}\end{array}\right]$

The given vectors are linearly dependent.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

Campbell University

Harvey Mudd College

Idaho State University

Lectures

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In this example, we have a matrix, a given here, which is of size three by four for three rows and four columns and what we want to determine, or whether or not the columns of a will be linearly, independent or dependent. To that end, consider the Matrix equation eight times X equals a zero vector. We want to determine if this equation has free variables or not. So ordinarily we might begin by row, reducing a. But this time we don't have to because we have size three by four, meaning we have more columns and rows. It's automatic that the columns of a are linearly dependent. Let's also explore why that has to be the case now. I don't know where the pivots are for this particular matrix, because I have not roll reduced it. But I do know that for the maximum amount of pivots, this would be our best case. Three columns are pivot columns, and that tells us that there is at least one free variable so automatically that implies that the columns of a forms a linearly dependent set and the solution is complete. So essentially, each time there is more columns than there are rows. It's automatic that in that particular matrix we'll have a linearly dependent set

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