# Solved: Find The Orthogonal Projection Of Y Onto U, Verify That

\$ u_1=egin{bmatrix}2 5-1 end{bmatrix}, u_2=egin{bmatrix}-2 11 end{bmatrix},y=egin{bmatrix}1 23 end{bmatrix}\$

I”m looking for the projection of \$y\$ on \$V^perp\$, knowing that \$V\$ is generated by \$S={u_1,u_2}\$.*

I did the projection of \$proj_V(y)\$:egin{align*}proj_V(y)&=frac{}{||u_1||^2}u_1+frac{}{||u _2||^2}u_2&=frac{3}{10}u_1+frac{1}{2}u_2end{align*}

But I don”t know how to find the projection of \$y\$ on \$V^perp\$.Do you have any hint? I don”t perefectly understand what \$V^perp\$ is…

linear-algebra orthogonality
Share
Cite
Follow
asked Nov 16 “15 at 13:45

Revolucion for MonicaRevolucion for Monica
\$endgroup\$

1
\$egingroup\$
The vector \$operatorname{proj}_V(y)\$ is the element of \$V\$ that is closest to \$y.\$ \$V^perp\$ is the set of all vectors \$v\$ such that \$langle u,v
angle=0\$ for all \$uin V\$, or (geometrically) the set of vectors that are orthogonal to every \$V\$-vector. Now, consider \$y-operatorname{proj}_V(y).\$

Share
Cite
Follow
answered Nov 16 “15 at 13:55

Cameron BuieCameron Buie
\$endgroup\$
0
\$egingroup\$
You can also simply find a (nonzero) vector in \$V ^ot\$ (how?), and then calculate the orthogonal projection of y onto this vector. Exercise:prove that the two methods are equivalent(you can think to the orthogonal decomposition of a vector onto a given direction).

Share
Cite
Follow
answered Mar 27 “16 at 20:32

Antonio Gabriele PalermoAntonio Gabriele Palermo
\$endgroup\$

Thanks for contributing an answer to wtbblue.comematics Stack Exchange!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

Use wtbblue.comJax to format equations. wtbblue.comJax reference.

Draft saved

Submit

### Post as a guest

Name
Email Required, but never shown

### Post as a guest

Name
Email

Required, but never shown

## Not the answer you're looking for? Browse other questions tagged linear-algebra orthogonality or ask your own question.

Featured on Meta
Visit chat
Related
1
Find orthonormal basis of \$wtbblue.combb{R}^3\$ with a given span of two basis vectors
1
Orthogonal Projections Help
1
Linear Transformation: Orthogonal Projections
2
The orthogonal projection of u onto v is 0?
1
Find the equation of an ellipse (isocontour )
1
How to prove the complement \$P^perp\$ of a projection matrix \$P\$ have relation \$I-P=P^perp\$
1
Find orthogonal projection of \$ ^T\$ on subspace \$V\$
0
Finding the projection of y onto the span of two vectors
Hot Network Questions more hot questions

Question feed