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Construct a $2 \times 2$ matrix $A$ such that the solution set of the equation $A \mathbf{x}=\mathbf{0}$ is the line in $\mathbb{R}^{2}$ through $(4,1)$ and the origin. Then, find a vector $\mathbf{b}$ in $\mathbb{R}^{2}$ such that the solution set of $A \mathbf{x}=\mathbf{b}$ is $n o t$ a line in $\mathbb{R}^{2}$ parallel to the solution set of $A \mathbf{x}=\mathbf{0} .$ Why does this not contradict Theorem 6$?$

see the solution

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

McMaster University

Baylor University

Lectures

01:32

In mathematics, the absolu…

01:11

03:19

Let $A$ be a $3 \times 4$ …

02:45

Use Theorem 3 (but not The…

Let $A$ be a $3 \times 2$ …

06:33

Suppose $A$ is an $n \time…

02:52

03:14

Explain why a $2 \times 2$…

01:29

Let $A$ be an $m \times n$…

03:10

Show that the equation of …

02:18

Suppose $A$ is a $4 \times…

02:12

Parallel line and plane Sh…

Okay. As part of the question, we have constructed by two metrics A such as the solution set off X equals tau zero is the line in our square through for one and the tourism. Okay. And the origin. And then we have to find a vector be in our squares as the resolution set off. A X equals toe B is not not aligning are two parallel to the solution. X equals 20 Okay, further, we have toe find that Is this Is this contacted from the Haram Six. So for this that we have a metrics toe by toe A, B, C and D. Okay, the question of the line through the region and the 60.41 Okay. What What is the Yeah, what are the point for one? So what will be the question of the line? That will be why it was 2.25 x. Okay. And that is any vector that satisfied this. This will be X 0.25 x. Okay, So what will be the metrics? That will be a B C D. Not ex toe 0.25 X equals toe zero and zero. Okay. And we can choose. For example, it was two Sequels to one, and B equals tau equals two minus four. And what we will get, We get one minus 41 minus four. Lord X 0.25 X equals two or this X minus for and to 2.25 eggs. This is X minus four and two 0.25 X. That will be 00 Okay for that. For this, the one off the possibilities is equals to one minus 41 minus four. Okay, so now we choose I'm Beaven and B two such that the equation. What will be the question? This will be one minus 41 minus four ex away. And the solution even. And we do. Okay. And this will be so it has no solution. Okay, because on this is enough to choose that. Be one. Not it was to be to, For example, even it was toe one and b two. It was 20 Then the question will be one minus 41 minus four. Don't x way and 10 Okay. And it has no solution. Okay, since it is impossible to say there some X off Satisfied? Both What X minus four Y questo one on X minus four by quest to zero. So the solution set off This equation is empty. Sad, And it is not the line parallel to vihk wants to 0.5 x. Understand? And further we have to see that is this tour. Um, is this solution contradict with Haram six. So we know that in the tour, um, six, we assume that a X equals Toby has a solution. Okay, in our example x, because Toby has no solution. Okay, so there is no contradiction. Put the two around result. Okay, so this will be our finally on the set, This one plus and this is from thank you.

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