Orthogonal Projections vs Non-orthogonal projections?

[rayzhu52]

Hi everyone,

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

Say for example, we are given the vectors:

y = [3, -1, 1, 13], v₁ = [1, -2, -1, 2] and v₂ = [-4, 1, 0, 3]

To find the projection of y, we first check is the set v₁ and v₂ are orthogonal:

v₁ • v₂ = -4 -2 + 0 + 6 = 0

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

\hat{y} =[(y • v₁)/(v₁ • v₁) * v₁)
+ [(y • v₂)/(v₂ • v₂) * v₂)]
= some value

Now, we covered what it means when a set is non-orthogonal v₁ • v_n≠ 0,
but what if we are asked to find \hat{y}?

Any form of help would be greatly appreciated!

 

[tiny-tim]

hi rayzhu52!

(btw, where did you get • from? use . or · (on a mac, it's alt-shift-9))

(and you've been using too many brackets )

the easiest way is probably to start by finding the unit normal, a multiple of v₁ x v₂