Specifically,
"Calculate the projection of the vector (1,2,3) onto the plane perpendicular to the vector (1,1,1)."
I am familiar with the projection formula, the projection vector $\mathbf v$ of $\mathbf w$,
$\mathbf v = P_u \,\mathbf w = A^T(AA^T)^{-1}A \,\mathbf w$, where the rows of matrix A form a basis for the subspace U. In this case, w = (1,2,3).
I cannot get my head around defining a subspace. From my understanding, the basis of a plane should need only two vectors. Please tell me if I'm wrong about this and why.
My question is:
- How do I define the matrix A based on the plane in the above question?
Thanks.
Best Answer
Caution: in the definition of $A$, it must the columns (and not the rows) of matrix $A$ ...
i.e., matrix $A$ has a rectangular shape ("portrait format" $3 \times 2$ and not "landscape format" $2 \times 3$) with its 2 columns equal to the basis on the space on which you project, here a plane.
You can take for example: $$A=\begin{pmatrix}\ \ 1 & \ \ 0\\ -1 & \ \ 1\\ \ \ 0 & -1 \end{pmatrix}$$
(check that the columns of $A$ are orthogonal to $\begin{pmatrix}1\\ 1\\ 1 \end{pmatrix}$)