Given and is a subspace,

Find the orthogonal projections of onto both and .

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Now, I've already found so I can decompose X into and s.t.

X = Y + Z.

Let Z = a(-6,-3,4,1), a in R.

So,

(I chose to use Z instead of Y since dim(Z)=1, but dim(Y)=3, so fewer inner-products.)

Dotting both sides with (-6,-3,4,1), we get:

since .

So and .

Y = X - Z so

Hope that's right!