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Suppose $a_1$ and $a_2$ are orthogonal unit vectors in $R^5$.

Where P is the projection matrix.

$P = a_1a_1^T + a_2a_2^T$

If $a_3$ is in $R^5$ , what combination of $a_1$ and $a_2$ is closest to $a_3$?

This has a simple solution. Just multiply matrix P with $a_3$

So we should have a result $a_1a_1^Ta_3 + a_2a_2^Ta_3$

But here the answer is given $a_1^Ta_3a_1 + a_2^Ta_3a_2$ . No process is stated though. Only the answer is given.

However $a^Ta$ equal to $aa^T$ as explained Product of a vector and its transpose (Projections)

Note when we are mentioning transpose we mean a vector in Matrix form. So I think that solution is incorrect. So my question is which solution is correct? And if the one given in the MIT link is correct ,then why?

Note: I know dot product is commutative but not it's matrix form.

Neer
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    Are you aware of the fact that $a_1^{T}a_3$ and $a_2^{T}a_3$ are scalars? So, actually there are some parentheses missing in the proposed solution. – trancelocation Jul 17 '20 at 05:55
  • @trancelocation wow, thank you. I got my answer. I didn't notice it. Before ,now I understand the two things are same. – Neer Jul 17 '20 at 06:16

1 Answers1

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As pointed to by @trancelocation, i didn't notice it before. Both the tings are same.

$a_1^Ta_3$ and $a_2^Ta_3$ are scalars.

So we have $a_1(a_1^Ta_3) + a_2(a_2^Ta_3)=(a_1^Ta_3)a_1 + (a_2^Ta_3)a_2$

Not deleting the question for future reference, if anyone else gets stuck.

Neer
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