# Thread: projection matrix

1. ## projection matrix

Let w = $\displaystyle \left(\begin{array}{ccc}-1\ \-1\ \-1\end{array}\right)$ Find the projection of w on space V.

$\displaystyle V =$ $\displaystyle <\left(\begin{array}{ccc} 1\ \ 4\ \ 0\end{array}\right)\left(\begin{array}{ccc} 1\ \ -1\ \ 2\end{array}\right)>$

I've already found $\displaystyle V^{perp}$ $\displaystyle =$ the span of $\displaystyle \left(\begin{array}{ccc} -4\ \ 1\ \ -1\end{array}\right)$

I worked out the projection matrix i.e. $\displaystyle A^T(A.A^T)^{-1}.A$ to be 3x3 matrix where $\displaystyle A = \left(\begin{array}{ccc} -4\ \ 1\ \ -1\end{array}\right)$

However i dont now where to bring w into the problem

2. Originally Posted by Tekken
Let w = $\displaystyle \left(\begin{array}{ccc}-1\ \-1\ \-1\end{array}\right)$$\displaystyle$ Find the projection of w on space V.

$\displaystyle V =$ $\displaystyle <\left(\begin{array}{ccc} 1\ \ 4\ \ 0\end{array}\right)\left(\begin{array}{ccc} 1\ \ -1\ \ 2\end{array}\right)>$

I've already found $\displaystyle V^{perp}$ $\displaystyle =$ the span of $\displaystyle \left(\begin{array}{ccc} -4\ \ 1\ \ -1\end{array}\right)$

This can't possibly be correct since $\displaystyle (1\,-\!\!1\,\,2)\cdot (-4\,\,1\,-\!\!1)\neq 0$ ....check your work: it must be $\displaystyle V^{\perp}=Span\{(-8\,2\,5)\}$

I worked out the projection matrix i.e. $\displaystyle A^T(A.A^T)^{-1}.A$ to be 3x3 matrix where $\displaystyle A = \left(\begin{array}{ccc} -4\ \ 1\ \ -1\end{array}\right)$

?? How a matrix with one single row is a 3x3 matrix? What did you actually mean here?

Tonio

However i dont now where to bring w into the problem
.

3. see below

4. Ooops sorry my bad, multiplied wrong

I wasn't saying the single row matrix was a 3x3 matrix... i was simply saying that i used the single row matrix to calculate the projection matrix (a 3x3) matrix using the formula

$\displaystyle A^T(A.A^T)^{-1}.A$

As my latex knowledge is still poor, it would have taken me too long to write out the 3x3 matrix...

Anyways thanks for pointing out my error in part 1 of the question