Gram-Schmidt orthogonalization

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   In the Gram-Schmidt process applied to $u = \begin{pmatrix} 0 \\ 20 \end{pmatrix}$ and $v = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$, compute the projection coefficient $\dfrac{u\cdot v}{v\cdot v}$ used to remove the $v$-component from $u$.

2. Warm-up

   In the Gram-Schmidt process applied to $u = \begin{pmatrix} 5 \\ 15 \end{pmatrix}$ and $v = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, compute the projection coefficient $\dfrac{u\cdot v}{v\cdot v}$ used to remove the $v$-component from $u$.

3. Warm-up

   In the Gram-Schmidt process applied to $u = \begin{pmatrix} 30 \\ 0 \end{pmatrix}$ and $v = \begin{pmatrix} 5 \\ -5 \end{pmatrix}$, compute the projection coefficient $\dfrac{u\cdot v}{v\cdot v}$ used to remove the $v$-component from $u$.

Problems shown under their original terms: Generated (template, solver-computed).