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need to find orthogonal projection of v on 2 diffrent w spaces $$ v= \begin{bmatrix} 2\\ 4\\ 6\\ 8\\ \end{bmatrix} $$

1.$$ w= span \begin{bmatrix} 1\\ 0\\ -1\\ 0\\ \end{bmatrix} \begin{bmatrix} 1\\ -1\\ 1\\ -1\\ \end{bmatrix} $$

2.$$ w= span \begin{bmatrix} 1\\ 0\\ 1\\ 0\\ \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix} $$

the answers should be vector like v (4 rows) thanks a lot.

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  • $\begingroup$ Please show your effort. $\endgroup$ Jun 23 at 6:12
  • $\begingroup$ idk what to do :( im sorry if it seems very basic, I have a few of t.hose questions with similar numbers like 0 0 1 -1 on last vector of w on q2. at this case for example to book said the answer is 6 0 2 4 $\endgroup$
    – Ghnb Trcv
    Jun 23 at 6:17
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    $\begingroup$ @GhnbTrcv There are a number of ways that a problem like this can be attacked, and some may not make sense to you depending on what you're learning currently. First, what book are you reading from? What kinds of topics have you been covering recently? Do words/phrases like "Gram-Schmidt", "Normal Equation", or even "Pseudoinverse" ring any bells? Let us know the context for this question, otherwise you might get an unhelpful answer, or more likely, have your question closed. $\endgroup$ Jun 23 at 7:00
  • $\begingroup$ yes Gram-Schmid is the thing I think $\endgroup$
    – Ghnb Trcv
    Jun 23 at 7:09
  • $\begingroup$ @GhnbTrcv Okay! So, next question, could you perform Gram-Schmidt on either of these sets of vectors? That would be the first step. $\endgroup$ Jun 23 at 7:12

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