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Let $v_{1} \ldots v_{k}$ be vectors in a linear space with an inner product $\langle,$,$rangle . Define the$ Gram determinant by $G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)$.
a) If the $v_{1} \ldots v_{k}$ are orthogonal, compute their Gram determinant.
b) Show that the $v_{1} \ldots v_{k}$ are linearly independent if and only if their Gram determinant is not zero.
c) Better yet, if the $v_{1} \ldots v_{k}$ are linearly independent, show that the symmetric matrix $\left(\left\langle v_{i}, v_{j}\right\rangle\right)$ is positive definite. In particular, the inequality $G\left(v_{1}, v_{2}\right) \geq 0$ is the Schwarz inequality.
d) Conversely, if $A$ is any $n \times n$ positive definite matrix, show that there are vectors $v_{1}, \ldots, v_{n}$ so that $A=\left(\left\langle v_{i}, v_{j}\right\rangle\right)$.
e) Let $\mathcal{S}$ denote the subspace spanned by the linearly independent vectors $w_{1} \ldots w_{k} .$ If $X$ is any vector, let $P_{\mathcal{S}} X$ be the orthogonal projection of $X$ into $\mathcal{S}$, prove that the distance $\left\|X-P_{\mathcal{S}} X\right\|$ from $X$ to $\mathcal{S}$ is given by the formula
$\left\|X-Z_{\mathcal{S}} X\right\|^{2}=\frac{G\left(X, w_{1}, \ldots, w_{k}\right)}{G\left(w_{1}, \ldots, w_{k}\right)} .$
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