Clumath2015s2ch6s3 Theorems

_Theorem 8_

The Orthogonal Decomposition Theorem

Let $W$ be a subspace of $\mathbb{R} ^ n$. Then each $y$ in $\mathbb{R} ^ n$ can be written uniquely in the form

(1)
\begin{align} y= \hat{y} +z \end{align}

where $\hat{y}$ is in $W$ and $z$ is in $W^\bot$. In face, if ${u_1 , . . . , u_p}$ is any orthogonal basis of $W$, then

(2)
\begin{align} \hat{y} = \frac{y\cdot u_1}{u_1 \cdot u_1 } u_1 + . . . + \frac {y \cdot u_p}{u_p \cdot u_p} u_p \end{align}

and $z=y-\hat{y}$

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