# 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
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asked Nov 16 “15 at 13:45

Revolucion for MonicaRevolucion for Monica
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## 2 Answers 2

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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).\$

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answered Nov 16 “15 at 13:55

Cameron BuieCameron Buie
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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).

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answered Mar 27 “16 at 20:32

Antonio Gabriele PalermoAntonio Gabriele Palermo
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