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Linear Algebra

Worksheet 15

  1. Apply Gram Schmidt to the following collection of vectors \(\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}\) By hand for this one, please.

  2. Apply Gram Schmidt to the following collection of vectors \(\begin{pmatrix} -2 \\ 3 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ -3 \\ -1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}\)

  3. Let $\langle -,- \rangle$ be a positive definite inner product on a vector space. Show that, if $w_1,\ldots,w_d$ is a set of nonzero orthogonal vectors, then it is a linearly independent. In particular, if $\dim V = d$, it is a basis.