Matrix - Vector Equations A system of linear equations can always be expressed in a matrix form. The matrix version of the equation has its own geometric interpretation. We think of a function which is defined for vectors by the following equation:

Showing that the candidate basis does span C A Video transcript In the last couple of videos, I already exposed you to the idea of a matrix, which is really just an array of numbers, usually a 2-dimensional array.

Actually it's always a 2-dimensional array for our purposes. So if I have an m by n matrix, the m is just the number of rows, and then the n is just the number of columns.

So let me write out the m by n matrix. So I'll just specify, let's have the m by n matrix A, it's a capital bold A. And it is equal to, I'll be as general as possible, first entry is in, I'll just call that lowercase a, it's in row 1 column 1. The next entry is row 1 column 2.

And you go all the way to row 1 column n, you have n columns. And then when you go down, you go to the next row, it will be row 2 column 1. And then you keep going all the way down to row m column n. And then of course, what? This entry is going to be, row 2, let me write that a little smaller, row 2 column 2.

And you go all the way, and you're going to have row m column n. And so if you think about it, you're going to have how many total entries here? You're going to have m entries this way, n that way.

So you're going to m times n total entries. And I think you're pretty familiar with this idea already of a matrix, you probably saw this in your Algebra II classes. So what we want to do now in this video is relate our notion of a matrix to everything we already know about vectors. Or maybe introduce some operations that allow matrix and vectors to interact with each other.

And maybe the most natural one is multiplication, or taking the product.

So what I'm going to do in this video is define what it means when we take the product of our matrix A, of any matrix A, I've written this as general as possible, with some vector x.

And our definition will only work if x, the vector we're multiplying A by, has the same number of components as A has columns. So this is only valid for an x that looks like this: So let me be very clear with this, this vector, I guess you could do it a different height than this vector.

What matters is that the same number of A's you have in this direction, you have n A's here, then you have n components of this vector right here.

And if you have that constraint, if the length of your vector, or the number of components in vector is equal to the number of columns in your matrix, then we define this product to be equal to -- so this is my vector x -- so this is a definition.

There's nothing in nature that told us it had to be defined this way.

It's just human beings, or mathematicians, decided that this is a useful convention to the define the multiplication, or the product, of a matrix and a vector.

So we'll define A times our vector x. These are both bold, this is a matrix, that's a vector. And the convention, if I didn't draw the little vector symbol, your textbooks would just bold out the x, so that it'll be a lowercase x.

Lower case is vector, uppercase is matrix, both of them are bolded. That tells you that you're not just dealing with regular numbers. So we're defining this to be equal to -- let me write it out fairly large.

You're going to take each row, and we're going to show you that there's multiple ways to kind of visualize this, but it's going to a11 times x1, let me write that down.

So a11 times x1 plus a12 times x2, all the way to plus a1n times xn. So the product of this matrix, this m by n matrix and this n component vector, will be a new vector, the first entry of which is essentially each of these entries times a corresponding entry here, and you add them all up.

And as you can see, that's already looking fairly similar to a dot product, and I'll discuss that in a second. But let me finish my definition before I start talking about what it means, or what it might be related to.

So that was that first row right there, it'll just look like that. We just multiply that times this thing to get that row there. Now the second row -- I want to do it in a different color -- remember this is a definition.Problem Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination).

General Solutions of Systems in Vector Form MA We are looking for solutions to the system Ax = b, in column vector form in what follows. We wish to organize the vectors We know that matrix multiplication is linear so we can check out the general solution as follows.

Here is the matrix I pulling the information from just in case you need it. \begin{bmatrix} 1 & 0 & -1\\ 0 & 1 & 1\\ 0 & 0 & 0 \end{bmatrix} This is what I am thinking.

\begin{bmatrix} 1\\ 0\\1 \end{bmatrix} \begin{bmatrix} 0\\ -1\\1 \end{bmatrix} This is my second general . Matrix - Vector Equations. You will choose the input vector by moving the mouse in the graph window. It will always be drawn as a red line from the origin to the point (x, y).

The output vector will be drawn in a similar fashion, always shown in blue.

The desired result,, will be shown in yellow. Nov 09, · This video shows an example of how to write the solution set of a system of linear equations in parametric vector form.

Vector notation is a commonly used mathematical notation for working with mathematical vectors, which may be geometric vectors or members of vector spaces.

For representing a vector, [5] [6] the common typographic convention is lower case, upright boldface type, as in .

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Matrix - Vector Equations