$A \in R^{m x n}$ matrix A of m rows and n cols
Inner Product: $x^Ty$ is dot product
Matrix-Vector Product: $y=Ax=\begin{bmatrix}a_1\\…\\a_n\end{bmatrix}x$
Matrix-Matrix Product: $C = AB = [a_1 … a_n]^T \cdot [b_1 … b_n]$
Consider system of linear equations, $Ax = b$ where $A$ is $n$ by $n$ and $x,b$ length $n$. If $A$ is invertible, then $x=A^{-1}b$.
Inverse is denoted as $A^{-1}$.
A square matrix $A$ has an inverse iff the determinant $|A| \neq 0$.
$A$ is invertible or non-singular if $A^{-1}$ exists and non-invertible or singular otherwise.
Calculating determinants:
$A = \begin{bmatrix}a_{11}&&a_{12}&&a_{13}\\a_{21}&&a_{22}&&a_{23}\\a_{31}&&a_{32}&&a_{33}\end{bmatrix} = a_{11}\cdot \mathrm{det} \begin{bmatrix}a_{22}&&a_{23}\\a_{32}&&a_{33}\end{bmatrix} - a_{12}\cdot \mathrm{det} \begin{bmatrix}a_{21}&&a_{23}\\a_{31}&&a_{33}\end{bmatrix} + a_{13}\cdot \mathrm{det} \begin{bmatrix}a_{21}&&a_{22}\\a_{31}&&a_{32}\end{bmatrix}$
A set of vectors $x_1,…,x_n$ is linearly independent if a sum of all of them multiplied by some constant equals 0 necessarily when all constants are 0.
Norm of a vector is informally a measure of the length of the vector.
$l_2$ norm: $||x||_2$ Euclidian norm. $||x||_2^2 = x^Tx$
Properties of norms: