# Finding the projection of a vector into a subspace

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 $v \in V$.

• 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 $v$? – The Count Nov 29 '16 at 3:57
• Do you know the Graham-Schmidt process to find the ortho-normal basis? – Doug M Nov 29 '16 at 3:59
• The Graham-Schmidt process converts a basis to an ortho-normal basis. – Doug M Nov 29 '16 at 4:00
• 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? – user198504 Nov 29 '16 at 4:02
• Doug i think your last comment is missing something – 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 $U$