Let $F = \mathbb{R}$ and $V$ be a vector space over $F$. Suppose $U$ is the subspace of $V$ spanned by $ \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} $ and $ \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix} $. Furthermore, $U$ has orthonormal basis $\{u_1, u_2\}$.

Find $\sum_{i=1}^2 \langle u_i, v \rangle u_i $ for any $v \in V$.

  • 1
    $\begingroup$ What do you know and what have you tried? Are your two column vectors supposed to be $u_1$ and $u_2$, respectively? What is $v$? $\endgroup$ – The Count Nov 29 '16 at 3:57
  • 1
    $\begingroup$ Do you know the Graham-Schmidt process to find the ortho-normal basis? $\endgroup$ – Doug M Nov 29 '16 at 3:59
  • $\begingroup$ The Graham-Schmidt process converts a basis to an ortho-normal basis. $\endgroup$ – Doug M Nov 29 '16 at 4:00
  • $\begingroup$ oh. so i use Grahm-schmidt to find the orthonormal basis based from a known basis which is the given spanning set? my question is i am not sure if the orthonormal basis unique. furthermore, is the projection unique? $\endgroup$ – user198504 Nov 29 '16 at 4:02
  • $\begingroup$ Doug i think your last comment is missing something $\endgroup$ – user198504 Nov 29 '16 at 4:04

$v_1,v_2 = \begin{bmatrix}1\\0\\-1 \end{bmatrix},\begin{bmatrix}0\\1\\-1 \end{bmatrix}$

Graham Schmid process.

find $v_2 - \frac {<v_1,v_2>}{\|v_1\|^2}v_1$

$\begin{bmatrix}-\frac {1}{2}\\1\\-\frac {1}{2} \end{bmatrix}$

Divide $v_1$ and this vector you have just found, each by its respective norm.

That is your basis $u_1, u_2$

$\sum <u_1, v> u_1 = v$ if $v\in U$ and is the projection onto $U$ if $v$ is not in $U$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy