Need serious help with orthogonal projections?? A linearly dependent set of vectors in R^4 is given. Find the standard matrix for the orthogonal projection of R^4 onto the subspace spanned by those vectors?

Question

anonymous · Answer

where a_1=(4, -6, 2 ,-4) a_2=(2,-3, 1,-2) a_3=(1,0,-2,5) a_4=(5,-6,0,1)

anonymous · Answer

to me it sounds like you want to create an orthogonal basis from those vectors give.  This requires the Gram-Schmit process.  Do you know about that, or are you familiar with it?

anonymous · Answer

Well I know what that is but that's the chapter after this so I don't think we use Gram Schmit process

anonymous · Answer

if its the chapter after it should be fair game no? I could understand if it was the chapter before....its like you havent learned it yet.  But after, it should be considered a tool for you to use.

anonymous · Answer

No haha I meant this question is the chapter before learning Gram Schmit, but yeah is that the quickest way then, I'm just studying for my final so whichever way is the quickest would be best

anonymous · Answer

im not too sure how you would do it without gram-schmit....i'll think about it for a bit lol.  but yeah that would be the most straightforward way to tackle this problem.

anonymous · Answer

Like if it was only one vector there's this formula (1/(A^T*A))*(A*A^T)

anonymous · Answer

and there is also this formula M(M^TM)^-1*M^T where M is any matrix. So I thought I could combine the vectors to make a matrix but not getting the answer

anonymous · Answer

Oh nvm I got it you do use the M formula, I put the vectors in rows but it's actually columns so it works now.

anonymous · Answer

hmmm, that sounds interesting, im gonna take a look at it too now, nice job!