$ u_1=eginbmatrix2 \5\-1 endbmatrix, u_2=eginbmatrix-2 \1\1 endbmatrix,y=eginbmatrix1 \2\3 endbmatrix$

I"m in search of the forecast of $y$ on $V^perp$, learning that $V$ is produced by $S=u_1,u_2$.*

I did the estimate of $proj_V(y)$:eginalign*proj_V(y)&=fracu_1+fracu_2\&=frac310u_1+frac12u_2endalign*

But I don"t recognize how to discover the forecast of $y$ on $V^perp$.Do you have any hint? I don"t perefectly understand what $V^perp$ is...

You watching: 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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The vector $operatornameproj_V(y)$ is the element of $V$ that is closest to $y.$ $V^perp$ is the collection of all vectors $v$ such that $langle u,v angle=0$ for all $uin V$, or (geometrically) the collection of vectors that are orthogonal to every $V$-vector. Now, take into consideration $y-operatornameproj_V(y).$

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answered Nov 16 "15 at 13:55

Cameron BuieCameron Buie

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You can likewise simply uncover a (nonzero) vector in $V ^ot$ (how?), and also then calculate the orthogonal projection of y onto this vector. Exercise:prove that the two approaches are equivalent(you deserve to think to the orthogonal decomposition of a vector onto a provided direction).

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answered Mar 27 "16 at 20:32

Antonio Gabriele PalermoAntonio Gabriele Palermo

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