The following statements are equivalent: | Square | Tall | Wide | | ----------------------------------------- | ------------------------------------------ | ------------------------------------- | | Linearly independent rows and columns | Linearly independant columns | Linearly independant rows | | $Ax = b$ is uniquely solvable for any $b$ | $Ax = b$ has a unique solution if solvable | $Ax = b$ is solvable for any real $b$ | | Invertible | Has left inverse | Has a right inverse | | ${} N(A)=0 {}$ | ${} N(A)=0 {}$ | | | ${} C(A)=\mathbb{R}^{n} {}$ | | ${} C(A) = \mathbb{R}^{m} {}$ | | ${} \text{Rank}(A)=m=n {}$ | ${} \text{Rank}(A)=n {}$ | $\text{Rank}(A)=m$ | **Upper Triangular Matrices** have $A_{i,j}=0$ for all $i<j$ **Lower Triangular Matrices** have $A_{i,j}=0$ for all $i>j$ **Rank-Nullity Theorem**: If $A$ is a $m$ x n matrix, $\underbrace{ \text{rank}(A) }_{ \text{pivots} }+\underbrace{ \text{dim}N(A) }_{ \text{free variables} }=n$ A non-negative matrix is a square matrix with only non-negative entries #### Permutation Matrices A permutation matrix is a square matrix with eactly one 1 in each row and column, and the rest is zeros. The following are true of permutation matrices: $A^{\mathrm{T}}= A^{-1}$ Doesn't change the length of a vector Doesn't change the angle between vectors