Permutation, transpose and space $R^n$

Permutations matrix P = Identity matrix with reordered rows.

$$ P^{-1} = P^T$$

$$P^TP = I$$

Transpose Matrix : $ (A^T)_{ij} = A_ji$ d

Symmetric matrix: $A^T = A $

$R^TR$ is always d

Why: $$(R^TR)^T = R^TR^{TT} = R^TR$$

$R^n$:

What do we do with vectors

  • add and substract
  • multiply by constant numbers

$R^2$ = all 2-D real vectors = x-y plane

Subspace of $R^2$

  1. all of $R^2$
  2. any line through $ \begin{bmatrix} 0 \\
    0 \end{bmatrix} $
  3. zero vector only

Columns in $R^3$

All their combations from a subspace called column space C(A)