Systems of Linear Equations
\begin{equation} T v = v' \end{equation}
every system of linear equations is decomposed into this. Classically, there’s either a unique solution, no solution, infinite solutions—
problems with zero
“zero” is really hard to define. For instance:
\begin{equation} 6.23423 \times 10^{192} - 1 \times 10^{7} = 6.23423 \times 10^{192} \end{equation}
so in this case \(10^{7}\) literally behaves like zero. (small numbers have the opposite problem)
so, we use elementary row operations to make sure that enormous numbers are essentially standardized—if a row has huge numbers, we may want to scale it down to smaller numbers to make them nice.
row scaling
scaling an entire row by multiplying the number a la elementary row operations
column scaling
scaling a column by changing the definition of \(c_{j}\); for instance,
\begin{equation} \mqty(3e-4 & 2 \\ 1e-4 & 0) \mqty(c_1 \\ c_2) = \mqty(\ddots) \end{equation}
we can set \(c_3 = c_1(1e-4)\) and write
\begin{equation} \mqty(3 & 2 \\ 1 & 0) \mqty(c_3 \\ c_2) = \mqty(\ddots) \end{equation}
the right side needn’t to get scaled since we simply changed the definition of \(x\).
square matrix
a square matrix is a invertable Linear Map.
solvability
singular matrix (non-solvable matrix) — see singular matrix:
- one column is linearly dependent on the others
- determinant is 0
- non-empty null space
Diagonal Matrix
see Diagonal Matrix