Determine the orthogonal complement of x in span [p1, p2, p3].

Asked Jun 03 '20 at 23:43

Active Jun 03 '20 at 23:43

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Let vectors P1, P2, and P3 be defined as follows:

P1 = [1 2 3 4] $^{T}$, P2 = [4 -2 -6 -7]$^{T}$, and P3 = [3 4 -2 1]$^{T}$

Let $x = [1 \, 2\, 3\, 7]^{T}$

This question is compromised into two parts:

a. Determine the closest vector $\hat{x}$ in span [P1, P2, P3].

b. Determine the orthogonal complement of x in span[P1, P2, P3].

Part a. is simply the projection x onto the subspace spanned by those vectors: $A(A^{T}A)^{-1}A^{T}x$. However, I am unsure about part b. I know that $x - \dot{x}$ is the orthogonal complement to the subspace spanned by [P1, P2, P3], but how do I find the orthogonal complement of x $\textbf{in}$ span [P1, P2, P3]?

asked Jun 03 '20 at 23:43

MRashid

Related – PinkyWay Jun 04 '20 at 00:14
You only have to take $x$ and normalize it and apply Gram-Schmidt further. – PinkyWay Jun 04 '20 at 00:17
The orthogonal complement is unique, so it should be straight-forward. – PinkyWay Jun 04 '20 at 00:18

Determine the orthogonal complement of x in span [p1, p2, p3].

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