线性代数导论 (7-Linear Transformations)
Linear Transformations
The Idea of a Linear Transformation
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 $w$ :
$$ T(cv+dw) = cT(v)+dT(w)$$
The Matrix of a Linear Transformation
$$\left {
\begin{array}{l}
\text{Linear transformation } T \
\text{Input basis } v_1,\ldots,v_n \
\text{Output basis } w_1,\ldots,w_m \
\end{array}
\right }
\Rightarrow
\begin{array}{c}
\text{Matrix } A (m\ by\ n) \
\text{represents T} \
\text{in these bases} \
\end{array}$$
Diagonalization and the Pseudoinverse
Polar Decomposition
The polar decomposition extends this factorization to matrices: orthogonal times semidefinite, $A = QH$.
$\textbf{Polar decomposition} \qquad A = UΣV^T=(UV^T)(VΣV^T)=(Q)(H)$
The Pseudoinverse
$$A^+= UΣ^+V^T$$
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).
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