$ 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…

Đang xem: Find the orthogonal projection of y onto u

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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$egingroup$

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 $y-operatorname{proj}_V(y).$

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answered Nov 16 “15 at 13:55

Cameron BuieCameron Buie

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$egingroup$

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).

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answered Mar 27 “16 at 20:32

Antonio Gabriele PalermoAntonio Gabriele Palermo

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