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:

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,

we can set c_3 = c_1(1e-4) and write

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