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My Linear Algebra Professor recently had a lecture on Orthogonal projections.

Say for example, we are given the vectors:

y = [3, -1, 1, 13], v

_{1}= [1, -2, -1, 2] and v

_{2}= [-4, 1, 0, 3]

To find the projection of y, we first check is the set v

_{1}and v

_{2}are orthogonal:

v

_{1}• v

_{2}= -4 -2 + 0 + 6 = 0

So we know the set is orthogonal and we can now find the projection of y, or [itex]\hat{y}[/itex]:

[itex]\hat{y}[/itex] =[(y • v

_{1})/(v

_{1}• v

_{1}) * v

_{1})

+ [(y • v

_{2})/(v

_{2}• v

_{2}) * v

_{2})]

= some value

Now, we covered what it means when a set is non-orthogonal v

_{1}• v

_{n}≠ 0,

but what if we are asked to find [itex]\hat{y}[/itex]?

Any form of help would be greatly appreciated!