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Suppose the last column of $A B$ is entirely zero but $B$ itself has no column of zeros. What can you say about the columns of $A ?$

$A$ is linearly dependent. Hence the columns of $A$ are linearly dependent.

Algebra

Chapter 2

Matrix Algebra

Section 1

Matrix Operations

Introduction to Matrices

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

02:05

Suppose the second column …

03:19

Suppose the first two colu…

01:17

Suppose the third column o…

01:05

By expanding the determina…

10:34

Show that if the columns o…

01:06

For what values of $a, b, …

00:30

Complete each statement. F…

02:41

Let $a, b,$ and $c$ be any…

00:28

Fill in the blanks.Amo…

02:19

Suppose the matrix represe…

in this example we have two major sees were considering a is of size and buy em B is of size and by P. We also have set the notation for the Matrix B so that columns are B one B two all the way through Bisa p With this information, if we were to say multiply a Times B, then by the definition of matrix multiplication, we would have a Times B one for the first column of the product. The second column is a Times B two all the way through a times B sub p for the last column of the product here. Let's suppose that a Times B s A P equals the zero vector and BCP itself is not the zero vector. This is an interesting statement, since a BCP is not a zero vector and we have a homogeneous equation we're considering. We can make an immediate conclusion here. This implies directly that the columns of the Matrix A are linearly dependent, and this is really through the definition of linear dependence. We have here an equation that could be converted into a vector equation, and we have a set of scale er's that come from Bisa P Not all equal to zero since we said this is not the zero vector And in this multiplication we have a linear combination set equal to zero with a nontrivial solution immediately calm survey are linearly dependent.

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