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Problem. Determine if it is true or false :

If the columns of a $n\times p$ matrix $U$ are orthonormal, then $UU^t y$ is the orthogonal projection of the vector $y$ onto the column space of $U$.

Hi, can someone tell me the theorem about this? I would like to Google it myself but i dont know how.

So someone who can explain me this or know what terms I have to Google for in Google?

Thank you!

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  • $\begingroup$ Please take the time to enter important parts of your question as text instead of pasting screen shots of it. Your question should be comprehensible with images disabled. Use MathJax to format mathematical expressions; you can find a quick reference here. $\endgroup$ – amd Jul 24 at 20:00
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If the columns of a matrix $A$ are independent, then projection onto $\operatorname{Col}(A)$ is given by $$ P_{\operatorname{Col}(A)}=A(A^\top A)^{-1}A^\top $$ Additionally, if the columns of $A$ are orthonormal, then $A^\top A=I$.

So, if the columns of $A$ are orthonormal, then projection onto $\operatorname{Col}(A)$ is given by $$ P_{\operatorname{Col}(A)} = A(A^\top A)^{-1}A^\top = AI^{-1}A^\top = AA^\top $$

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