linear-algebra-notes

Matrix factorization

Equation	Purpose
$A=LU$	Elimination
$A=QR$	Gram-Schmidt orthogonalization
$S = Q\Lambda Q^T$	$\Lambda$ eigenvalues, $Q$ eigenvectors
$A = X\Lambda X^{-1}$
$A = U\Sigma V^T$	SVD

Four fundamental subspaces

For matrix $A, m \times n$ , rank $k$

Column space, $C(A)$
Row space, $C(A^T)$
Null space, $N(A)$
Null space, $N(A^T)$

Dimensionality of column and row space is $k$ , which is significant as it means even for large rectangular matrices, the space that columns and rows span is constant.

The reason why this is significant there are a finite number of solutions to linear equations
Null space is all solutions $Ax = 0$
Dimension of null space is $\text{dim} = n - k$
- $n$ variables, $k$ constraints