Let $$\mathbf{v}_1 = \left[\begin{array}{c}1\\2\\0\\-3\\0\\3\\\end{array}\right], \mathbf{v}_2 = \left[\begin{array}{c}0\\0\\5\\-3\\0\\2\\\end{array}\right], \text{ and } \mathbf{v}_3 = \left[\begin{array}{c}1\\0\\0\\-1\\3\\2\\\end{array}\right].$$ Use the Gram-Schmidt procedure to produce an orthogonal set with the same span. (Hint: If you have unlimited submissions, it might be useful to submit to check your answer for each vector $\mathbf{u}_j$ before continuing.) Note: The $\mathbf{u}_j$ must be given in the same order as provided by the standard procedure.

$\mathbf{u}_1 =$ $\left[\Rule{0pt}{7.2em}{0pt}\right.$$\left]\Rule{0pt}{7.2em}{0pt}\right.$

$\langle \mathbf{u}_1, \mathbf{v}_2 \rangle =$

$\langle \mathbf{u}_1, \mathbf{u}_1 \rangle =$

$\mathbf{u}_2 =$ $\left[\Rule{0pt}{7.2em}{0pt}\right.$$\left]\Rule{0pt}{7.2em}{0pt}\right.$

$\langle \mathbf{u}_1, \mathbf{v}_3 \rangle =$

$\langle \mathbf{u}_2, \mathbf{v}_3 \rangle =$

$\langle \mathbf{u}_2, \mathbf{u}_2 \rangle =$

$\mathbf{u}_3 =$ $\left[\Rule{0pt}{7.2em}{0pt}\right.$$\left]\Rule{0pt}{7.2em}{0pt}\right.$