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Let $\mathbf{q}_{1}, \mathbf{q}_{2}, \mathbf{q}_{3},$ and $\mathbf{v}$ represent vectors in $\mathbb{R}^{5},$ and let $x_{1}, x_{2}$ and $x_{3}$ denote scalars. Write the following vector equation as a matrix equation. Identify any symbols you choose to use. $x_{1} \mathbf{q}_{1}+x_{2} \mathbf{q}_{2}+x_{3} \mathbf{q}_{3}=\mathbf{v}$

$q_{1}=\left[\begin{array}{l}{q_{11}} \\ {q_{12}} \\ {q_{13}} \\ {q_{14}} \\ {q_{15}}\end{array}\right], q_{2}=\left[\begin{array}{l}{q_{21}} \\ {q_{23}} \\ {q_{24}} \\ {q_{25}}\end{array}\right], q_{3}=\left[\begin{array}{l}{q_{31}} \\ {q_{32}} \\ {q_{34}} \\ {q_{35}}\end{array}\right]$ and $v=\left[\begin{array}{c}{v_{1}} \\ {v_{2}} \\ {v_{3}} \\ {v_{4}} \\ {v_{5}}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 4

The Matrix Equation Ax D b

Introduction to Matrices

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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in this example, we're going to start off with three different vectors. Q. One Q 23 Q 3 which come from our five and then three different scale. ER's X one through X three, which are from the set of real numbers are with these components. We can write down the following following a vector equation. It'll read x one times Q one plus x two times Q two plus x three times Q three, which results in a new vector V and that vector V has to come from our five once again. So this is our vector equation, and what we would like to do in this video is re expressed the vector equation as a matrix equation. So the best way to express this matrix equation isas compactly as possible. It's going to be of the form a Times X as a vector equals V, so the right hand side equals V doesn't have to change, but we're going to express what the A and the X have to be. So we have one more step. First, let's write in what the Matrix A is. I remember The Matrix say is going to contain the vectors that came from the vector equation. They come from our five, and we could let these be. Q one Que two and Q three written this way. It says that our Matrix A has first column equal to Q one and so on. Now let's work with the ex portion of the Matrix equation. We have a Times X and there are three scales for X. They are x one, x two and x three. Then the whole thing is set equal to the vector V. So this is our matrix equation, and our solution is complete, but still a good idea to do one extra check. The matrix that we listed here is going to be of size five, which comes from the fact the vectors Aaron are five by three, which comes from the fact that there are three vectors here. Next. We also have that this vector is of size three by one, and that tells us we're in good shape since eight times X is defined Onley because thes two dimensions highlighted in yellow match. So if we did some kind of work, where say, maybe there's a four here instead of a three, this check will allow us to detect if there is a problem

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