Linear Relations and Matrix Multiplication

What Are Matrices?

A matrix is a rectangular array of numbers or other data. In mathematics, it is a rectangular
array of numbers whose elements are indexed by non-negative integers. Although the
term "matrix" can refer to diverse mathematical objects, its most familiar meaning is that of
an n-by-n table filled with elements whose ith entry is denoted by aij.
So maybe you have had a little exposure to matrices in high school, but if you haven't,
here is just enough background so that you can follow along with the material.

If you want
to know more about the life and times of matrices, then by all means take 1806 sometime.
In life, many phenomena are related by linear formulas. Even when they are not and
cannot be approximated by linear formulas, in some cases you can use linear formulas to
approximate them anyway.
So often we find linear relations between variables. And for example, if we do a change of
coordinate systems. So for example, say that we are in space. And we have a point. Its
coordinates in my initial coordinate system might be x1, x2, x3.
But if I switch to a different coordinate system. I may be better able to solve my problem
because the new coordinates are more suited to other tasks I will perform in my solution
In these new coordinates, I will have other coordinate values-u1, u2 and u3.
And then, if I choose the same origin, the relation between the old and the new
coordinates will be linear.

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I mean, especially if I choose different origins, otherwise there
might be constant terms which I won't insist on. So let me just give an example.
Suppose, for example, that u1 equals twice x1 plus 3 times ×2 plus 3 times x3. U2 might
equal 2 times x1 plus 4 times ×2 plus 5 times x3. U3 might equal x1 plus x2 plus 2×3.
Numbers come from nowhere, they are made up.
OK, that's just an example of what might happen. So, if you want to express this kind of
linear relation in matrices, we can use matrix multiplication. So a matrix is just a table with
numbers in it and we can reformulate this in terms of matrix product.
OK, so instead of writing this, I will write that for matrix 2, 3, 3; 2, 4, 5; 1, 1, 2 times the
vector x1, x2, ×3 is equal to u1, u2 and u3. Hopefully you see that there is the same
information content on both sides. I need to explain to you what this way of multiplying
tables of numbers means: we will take exactly these quantities.
However, I would like to express this more symbolically. Let us call the first matrix A and
the second matrix X. Then we could say that A times X equals U
I need to explain what A, X, and U are, so that you can understand their entries in the
formula. But this is convenient notation.
The entries in a matrix product are obtained by dotting each row vector with its
corresponding column vector to obtain new row vectors. So, for example, we can compute
the product A X by dotting the first row of A with the first column of X.
We can perform the dot product between the rows of A and the columns of X. A is a 3×3
matrix, meaning that there are three rows and three columns in A. We can think of X as a
3-element column vector, or a 1-by-3 matrix. It has three rows and only one column.
Now, what can we do? As said, we would calculate the dot product between a row of A–2.
3, 3 and a column of X–xl, x2, x3.
The dot-product of two vectors, x1 and ×2, results in 2×1 plus 3×2 and 3×3. Since this is
what we want to set ul equal to, we are done with this step.
Let's do the second one. I take the second row of A-2. 4. 5 and multiply it by x1x2x3. I get
2×1 plus 4×2 plus 5×3, which is this thing. And same with the third one. 1×1 plus 1×2 plus
2×3. OK, so that was matrix multiplication.
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