Matrices are more than just tools for finding solutions to these systems of equations: they are rich mathematical objects in their own right, and it is the study of them that defines linear algebra.

Thoughts: One of the great things about Mathematics is how things get built upon each other, into wildly new fields, yet retaining fundamental concepts, and being relatable to other field: $c$ and $d$ are functions of $ax + by$ and $ax + by + cz$ respectively, and this also relates to the function focused aspect of the fundamentals of calculus, in that we can say $y = f(x) = ax+by+cz$. Also, we know that we are dealing with real numbers, due to the nature of the coordinate system not being complex, so we can assume that $a,b,c,x,y,z \in \mathcal{R}$, which is the set of real numbers for which they have membership, so there are even reminisces of set theory hidden within this. This makes sense though, because the foundations of mathematics have been set theoretic since the early 1900's.

$a_1x_1 + a_2x+2+ \ldots a_nx_n = b$.

A homogeneous linear equation is one where $b = 0$.

Within this class of equation, there are no products or roots of variables, and said variables only appear in the first power.

Within a system of linear equations, the variables are the unknowns of the system, with the values being coefficients.

$\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n = b_1 \ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_{n} = b_2 \ \vdots \ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n = b_m \end{array}$

For instance the equations of the system $\begin{array}{lcl} a_1x +b_1y = c_1 \ a_2x + b_2y = c_2 \end{array},$ when graphed, are lines, and so each solution ($x,y$) corresponds to a point of intersection, so there are three possibilities:

$\begin{array}{lcl} a_1x +b_1y + c_1z = d_1 \ a_2x + b_2y + c_2z = d_2 \ a_3x + b_3y + c_3z = d_3 \end{array}$

While these planes can relate to each other in six different ways now, the solution(s) of the system are still restricted to the three types of possibilities:

$\begin{bmatrix} a_{11}\; a_{12} \ldots a_{1n}\; b_1 \ a_{21} \; a_{22} \; \ldots a_{2n} \; b_2 \ \vdots \ a_{m1}\; a_{m2} \ldots a_{mn}\;b_m \end{bmatrix}$

This is the augmented matrix for the system.

Multiplying each term of an equation by a constant.

Interchanging two equations.

Multiplying each term of an equation by a constant and adding it to another equation.

a.) $x_1 +5x_2 − \sqrt{2}x_3 =1$.

b.) $x_1 + 3x_1 + x_1x_3 = 2$

c.) $x_1 = -7x_2 + 3x_3$

d.) $x^{\scriptsize{-2}}_1 +x_2 +8x_3 = 5$

e.) $x^{3/5}_1 - 2x_2 + x_3 = 4$

f.) $\pi x_1 - \sqrt{2} x_2 = 7^{1/3}$

a.) $2^{1/3}x + \sqrt{3}y = 1$

b.) $2x^{1/3} + 3 \sqrt{y} = 1$

c.) $cos(\frac{\pi}{7})x - 4y = log3$

d.) $\frac{\pi}{7}cos x - 4y = 0$

e.) $xy = 1$

f.) $y + 7 = x$.

a.) two equations in two unknowns:

b.) three equations in three unknowns

c.) two equations in four unknowns

a.) $\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$

b.) $\begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$

c.) $\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \ a_{21} & a_{22} & a_{23} & a_{24} \end{bmatrix}$

a1.) $\begin{bmatrix} 2 & 0 & 0 \ 3 & -4 & 0 \ 0 & 1 & 1 \end{bmatrix}$

b1.) $\begin{bmatrix} 3 & 0 & -2 & 5 \ 7 & 1 & 4 & -3 \ 0 & -2 & 1 & 7 \end{bmatrix}$

a2.) $\begin{bmatrix} 0 & 3 & -1 & -1 & -1 \ 5 & 2 & 0 & -3 & -6 \end{bmatrix}$

b2.) $\begin{bmatrix} 3 & 0 & 1 & -4 & 3 \ -4 & 0 & 4 & 1 & -3 \ -1 & 3 & 0 & -2 & -9 \ 0 & 0 & 0 & -1 & -2 \end{bmatrix}$

a1.)

b1.)

c1.)

a2.) $3x_1 - 2x_2 = -1 \ \hspace{2em} 4x_1 + 5x_2 \hspace{0.3em}= 3 \ \hspace{2em} 7x_1 + 3x_2 \hspace{0.3em}= 2$

b2.)$\; 2x_1 \hspace{2em} +2x_3 = 1 \ \hspace{2.2em} 3x_1 - x_2 + 4x_3 = 7 \ \hspace{2.2em} 6x_1 + x_2 - x_3 \hspace{0.5em}= 0$

c2.) $x_1 = 1 \ \hspace{2.4em} x_2 = 1 \ \hspace{2.4em} x_3 = 1$