Question

Give the Projection of a vector onto a subspace

Question:
https://imgur.com/Tj3dUiC

Totally confused on where to start with this, anyone able to help me understand how to answer this question?

Answer 1

This link seems to have a few examples: http://www.cliffsnotes.com/math/algebra/linear-algebra/real-euclidean-vector-spaces/projection-onto-a-subspace

Answer 2

So to start, you have
\[\begin{align*}\large \text{proj}_UX&=\large\sum_{i=1}^3\text{proj}_{X_i}X\\\\
&=\large\text{proj}_{X_1}X+\text{proj}_{X_2}X+\text{proj}_{X_3}X\\\\
&=\frac{X\cdot X_1}{X_1\cdot X_1}X_1+\frac{X\cdot X_2}{X_2\cdot X_2}X_2+\frac{X\cdot X_3}{X_3\cdot X_3}X_3
\end{align*}\]
\[X\cdot X_1=(3,1,0,42)\cdot(1,0,-1,-1)=-39\\
X\cdot X_2=(3,1,0,42)\cdot(2,1,1,1)=49\\
X\cdot X_3=(3,1,0,42)\cdot(-1,3,-1,0)=0\]
So,
\[\text{proj}_UX=-39X_1+49X_2=\begin{bmatrix}59\\49\\88\\88\end{bmatrix}\]

Answer 3

You want the orthogonal projection, which is given by
\[X_{\perp U}=X-\text{proj}_UX\]
and that's
\[X_{\perp U}=\begin{bmatrix}3\\1\\0\\42\end{bmatrix}-\begin{bmatrix}59\\49\\88\\88\end{bmatrix}=\begin{bmatrix}-56\\-48\\-88\\-46\end{bmatrix}\]
Hmm, that doesn't look entirely right to me...

Answer 4

So I got,

Based on the Best approximation theorm,

Where,

\[Proj_UX = \frac{X·X_1}{X_1·X_1}X_1 + \frac{X·X_2}{X_2·X_2}X_2 + \frac{X·X_3}{X_3·X_3}X_3\]

\[\left(\begin{matrix}1 \\ 7\\ 20\\ 20\end{matrix}\right) = Proj_UX\]

Does this seem correct?

Answer 5

Here is the theorem

Answer 6

Also I computed the error as,

\[||X - proj_uX|| = \sqrt{924}\]

Answer 7

Hmm, I'm afraid I couldn't say one way or the other. My memory of linear algebra is fading a lot faster than I'd anticipated. In any case, I don't think I've ever seen that theorem before. Sorry I can't help :/

Answer 8

Its alright, I looked it up and found a textbook that gave an example. Thanks anyways :)