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Let $A$ be an $m \times n$ matrix, and let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^{n}$ with the property that $A \mathbf{u}=\mathbf{0}$ and $A \mathbf{v}=\mathbf{0} .$ Explain why $A(\mathbf{u}+\mathbf{v})$ must be the zero vector. Then explain why $A(c \mathbf{u}+d \mathbf{v})=\mathbf{0}$ for each pair of scalars $c$ and $d$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:46

Let $A$ be an $m \times n$…

01:54

For the arbitrary vectors …

01:57

02:10

01:52

Let $\mathcal{B}=\left\{\m…

02:21

00:59

03:12

00:34

Let $\mathbf{u}=\langle a,…

00:29

All right. So in this problem, work even eight times you equals zero and the times be it was zero. Now, first we need to figure out eight times. You plus B. Now, this is quite simple. We just We know that a matrix matrix multiplication is distributed so unjust that apply the law, which is eight times you plus 800 now. It had serious 08 times v zero. Also, it is zero now. Secondly, we need Thio. Find out many times. See Time's Hugh plus the Times V Now C and D are skaters. So again we apply the, um it's reputed ity. So that's 80 times. See, do you plus eight times Dean. Yeah. Now, since Cmdr scale er's, we can take the skater impromptu off the expression so that it's C times. Hey, you last e kinds a V. Now, right now, I can apply this associative ity off matrix multiplication so that it's C times zero us d times zero. So that is zero and

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