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Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$.

Find the orthogonal projection of $w$ onto V,using exact values in your answer.


My Approach

Let the orthogonal projection be $s$

I used the formula:

$s=(w\cdot v_1)v_1+(w\cdot v_2)v_2$

Where $v_1,v_2$ are the vectors that span V.

But it turns out that I am wrong. What am I doing wrong?

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Your basis vectors need to be normalized for your formula to work. Divide each of your spanning vectors by its own magnitude, and use the resulting unit vectors as your v1 and v2. Then your formula will work.

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Above formula works only if you have orthonormalised the vectors $v_1,v_2$.

To get orthogonal projection of a vector on a subspace, first find basis of subspace say it is $\{{v_1,v_2,...v_m}\}$

Now using Gram-schimdt orthonormaization orthonormalize above one. Say it is $\{{w_1,w_2,...w_m}\}$.

$s=(w\cdot w_1)w_1+(w\cdot w_2)w_2+....+(w\cdot w_m)w_m$ is your required projection.

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  • $\begingroup$ His v1 and v2 are orthogonal. $\endgroup$ Aug 31 '15 at 6:51
  • $\begingroup$ right, but are not "orthonormal' $\endgroup$ Aug 31 '15 at 6:52
  • $\begingroup$ May I suggest you add the general formula in your answer? That could be helpful. $\endgroup$ Aug 31 '15 at 7:02
  • $\begingroup$ @ Thibaut Dumont, edited. $\endgroup$ Aug 31 '15 at 7:16

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