Questions

01

Orthogonal vectors, finding components

Show that the set of vectors is orthogonal:

$${\mathbf{u}}_1=\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right),{\mathbf{u}}_2=\left(\begin{array}{c}-1\\ 1\\ 2\end{array}\right),{\mathbf{u}}_3=\left(\begin{array}{c}2\\ -2\\ 2\end{array}\right),$$

and then compute the components of $\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$ in the basis $\{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\}$.

02

Perpendicularity to a subspace

Explain and justify in more detail the last statement: “${\mathbf{u}}_i\cdot \mathbf{x}=0$ for all $i$” implies “$\mathbf{u}\cdot \mathbf{x}=0$ for all $\mathbf{u}\in U$.”

03

Dependence?

What happens in the above Gram-Schmidt process if $\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is a dependent set?