Gram Schmidt¶
Gram Schmidt
Let \(V\) be a finite dimensional inner product space and \(\{v_1,\dots,v_k\}\) be a linearly independent subset of \(V\). The vectors \(w_i\) recursively defined by
\[
\begin{cases}
w_1=v_1\\
w_j=v_j-\displaystyle\sum_{i=1}^{j-1}\frac{\langle v_j,w_{i}\rangle}{\|w_{i}\|^2}w_{i},&\text{if }2\le j\le k
\end{cases}
\]
form an linearly independent and orthogonal set. Moreover, the span of \(\{v_1,\dots,v_k\}\) is equal to the span of \(\{w_1,\dots,w_k\}\).
Quote
Gram Schmidt 就是把下一個向量扣掉關於排在他前面的所有向量的正射影，得到的其實就是垂直分量。