The Gram-Schmidt orthogonalization procedure

In $\mathbb{R}^{3}$ with the usual dot product, find an orthogonal basis for

In $\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for

In $\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for

Let

and consider the vector space $\mathbb{R}^{3}$ with the inner product given by $\langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{T}A\mathbf{w}$. Let

Apply the Gram-Schmidt procedure to $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$ for $\mathbb{R}^{3}$ with respect to the above inner product.

Let

and consider the vector space $\mathbb{R}^{4}$ with the inner product given by $\langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{T}A\mathbf{w}$. Let

and let $W=\operatorname{span}\left\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right\}$. Apply the Gram-Schmidt procedure to $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$ for $W$ with respect to the above inner product.

Consider the inner product space $C[0,2]$, with the inner product given by

Use the Gram-Schmidt procedure to find an orthogonal basis for $\operatorname{span}\left\{1,x,x^{2}\right\}$.

Find an orthonormal basis for the subspace of $\mathbb{R}^{3}$ from Exercise 13.