Llamas
Hello! This is my first real post, so this is going to mostly just be a test of my Markdown abilities.
Anyways I thought it was interesting how repetitive the introduction of linear algebra can be. That’s not to say that the repetition is bad, it’s just that we’ve found a lot of ways to essentially say the same thing.
Consider a system of $m$ equations with $n$ variables:
\[\begin{gather*} a_{11}x_1 + a_{12}x_2 + a_{13}x_3 + \cdots + a_{1m}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + \cdots + a_{2m}x_n = b_2 \\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + \cdots + a_{3m}x_n = b_3 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + a_{m3}x_3 + \cdots + a_{mn}x_n = b_m \end{gather*}\]Even if we had no idea how to do Linear Algebra, we could probably solve for $x_1, x_2, x_3 \cdots$. (provided that a solution exists, of course). We would just use a combination of elimination and substitution.
Augmented Matrices
But mathematicians got lazy and were looking for easier way to solve them. So they took all the coefficients ($a_1, a_2, a_3 \cdots$) of the variables ($x_1, x_2, x_3, \cdots$) in the equations and made a matrix, like so:
\[\begin{equation*} A_{m,n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \end{equation*}\]And then they just appended the numbers $b_1, b_2, b_3$ to the end. We’ve obtained what is known as the augmented matrix!
\[\begin{equation*} A_{m,n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & b_{1}\\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{bmatrix} \end{equation*}\]Now you can do a bunch of row operations and solve the equations.
Vector Equations
In fact, the augmented matrix and vector equations are exactly equivalent forms! You can prove it yourself, with some very straightforward expansion.
\[\begin{align*} x_1 \begin{bmatrix} a_{11} \\ \vdots \\ a_{m1} \end{bmatrix} + x_2 \begin{bmatrix} a_{12} \\ \vdots \\ a_{m2} \end{bmatrix} + x_3 \begin{bmatrix} a_{1m} \\ \vdots \\ a_{mn} \end{bmatrix} = \begin{bmatrix} b_{1} \\ \vdots \\ b_{m} \end{bmatrix} \end{align*}\]Matrix-Vector Multiplication
And all this, in fact, is equal to matrix-vector multiplication. I won’t get into the long proof here because it’s annoying to write out and boring. If you know matrix-vector multiplication you should be able to prove it for yourself.
\[\begin{align*} A\vec{x} = \vec{b} \end{align*}\]A is the coefficient matrix, $\vec{x}$ is a column vector of all your variables and $\vec{b}$ stores the solutions to each equation. Linear systems of equations, augmented matrices, and vector equations and matrix-vector multiplication are all basically the same thing. Alright, that’s all for now. No new thoughts, just figuring out how to write.
Cheers.
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