Linear Algebra Question?

Question

Hi,

I have a question regarding Orthogonal sets and projections.

Say we have the vectors y = [3, -1, 1, 13],  v(1) = [1, -2, -1, 2] and v(2) = [-4, 1, 0, 3]

The projection of y, or y hat, would be found using the equation:

y hat = (y • v(1))/(v(1) • v(1)) * (v(1)) + (y • v(2))/(v(2) • v(2)) * (v(2))

IF the set is orthogonal.

But what do we do if it isn't orthogonal? How would we find the projection of y then?

I read through my textbook today but cannot seem to find what I'm looking for...

Any help would be greatly appreciated!

Indica · Accepted Answer

Here are two ways    …

Let V = span(v₁,v₂). You want the projection of y onto the subspace V.

(i) Find an orthogonal basis for V, say u₁,u₂. Do this is using Gram-Schmidt method. For a pair of vectors this is particularly easy   ���  u₁=v₁, u₂=v₂−(v₂•u₁)u₁/|u₁|²

Then carry on as before.

(ii) Assemble the projection matrix P(4X4) so then the projection of y is Py.

Find P using P = A(AᵀA)⁻¹Aᵀ where A(4X2) is matrix with columns v₁,v₂ (quotable result)

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