线性代数导论 (7-linear transformations)

A transformation $T$ assigns an output of $T(v)$ to each input vector $v$ in $V$. The transformation is linear if it meets these requirements for all $v$ and $w$ :
$$ T(cv+dw) = cT(v)+dT(w)$$

$$left {
begin{array}{l}
text{Linear transformation } T
text{Input basis } v_1,ldots,v_n
text{Output basis } w_1,ldots,w_m
end{array}
right }
Rightarrow
begin{array}{c}
text{Matrix } A (m by n)
text{represents T}
text{in these bases}
end{array}$$

The polar decomposition extends this factorization to matrices: orthogonal times semidefinite, $A = QH$.

$textbf{Polar decomposition} qquad A = UΣV^T=(UV^T)(VΣV^T)=(Q)(H)$

$$A^+= UΣ^+V^T$$

The pseudoinverse $A$ transforms the column space of $A$ back to its row space. $A^+A$ is the identity on the row space (and zero on the nullspace).