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Prove the second part of Theorem $6 :$ Let $\mathbf{w}$ be any solution of $A \mathbf{x}=\mathbf{b},$ and define $\mathbf{v}_{h}=\mathbf{w}-\mathbf{p} .$ Show that $\mathbf{v}_{h}$ is a solutionof $A \mathbf{x}=\mathbf{0} .$ This shows that every solution of $A \mathbf{x}=\mathbf{b}$ has theform $\mathbf{w}=\mathbf{p}+\mathbf{v}_{h},$ with $\mathbf{p}$ a particular solution of $A \mathbf{x}=\mathbf{b}$ and $\mathbf{v}_{h}$ a solution of $A \mathbf{x}=\mathbf{0} .$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

Campbell University

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:32

In mathematics, the absolu…

01:11

03:13

Prove that if $\mathbf{x}_…

00:16

What is the first step to …

07:04

Complete the two-column pr…

04:47

Let $P(x)$ be a polynomial…

05:02

Prove Property 6 of Theore…

03:57

Prove Formula 6 of Theorem…

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Give the reasons for each …

03:59

03:06

Complete the paragraph pro…

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Show that the product of t…

okay in discussion. What has given his first question? Question X equals Toby is consistent for some given. Be okay. And a letter PBS solution off this. Okay. And further vector on this is the effect, er people us v edge on for their V edge is any solution for the homogeneous equation. X equals 20 Okay, so these are given and what we have to prove. We have to prove two things. First, any vector off the form. Yeah, people, us. We ash is the solution. Two x equals to be okay. And second thing, we have to prove that every solution toe X equals Toby has deformed p. Plus we h Okay, these two things we have to prove. Okay, so now we have now we will do the first proof. Okay, So the compute the given question is X was Toby for that? He is always a solution. Okay. Solution. So what? We write a X equals Toby for their We can write that a people us we edge. Okay. For solving this, we get a P plus a V edge. And for the discussion about we know that a P p is the solution off e x x equals Toby. Okay, so this will be be here. And V Edge Vector has a solution off a X equals to zero. So this will be zero. So this will be be okay, so we get a P plus we edge. It was to be so every vector of the form people us, we edge people. Us. We're satisfied. The equation a x plus b. Okay, so this will be the proof off Approval one. Okay, now we have to prove second and the second waas second that we have to prove that every solution, every solution toe X equals Toby has deformed has the form p plus we h okay. It's nothing so difficult. So let the blue be a solution. Let the blue B ISS solution toe a X equals Toby. Okay, now no, on w will be b plus something. Ok, we have to find this something. So on Wayne always that the blue minus p is a solution off homogeneous equation X equals 20 Okay, so when we saw a the blue minus p equals two for elaborated, it will be a w minus a p. Okay. And now the interesting point comes here that a w the W is the solution off X equals toe Be okay. W is the solution off a X equals Toby. So it will be be now, and B is earlier. We have brought that p is also solution off X equals Toby. So this will be also be so this will be zero hair. Okay, so 40 so we can write that w has the form p plus something. And what is something? It is the blue minus b. Okay, so, Doug, end the blue minus b is a solution. Toe the homogeneous equation X equals two zero. Okay, So in result, we can write as a V h equals toe a w minus B. That is a no loo minus a B. That is gentle. Thank you.

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