Find the projection of a vector onto a subspace of $\Bbb R^4$

Question

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as:

$$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$

Thanks for any help i get guys.

Answer 1

First find a basis for $V$. We have $$\eqalign{V &=\{\,(x,y,2x-y,x-y)\mid x,y\in{\Bbb R}\,\}\cr &=\mathop{\rm span}\{\,(1,0,2,1),\,(0,1,-1,-1)\,\}\ ,\cr}$$ and so $$\def\v#1{{\bf #1}} \v w_1=(1,0,2,1)\ ,\quad \v w_2=(0,1,-1,-1)$$ form a basis for $V$. Now the projection of $\v b$ onto $V$ is $$\lambda_1\v w_1+\lambda_2\v w_2\ ,\tag{$*$}$$ where $$\v b=\lambda_1\v w_1+\lambda_2\v w_2+\v w\tag{$*{*}$}$$ and $\v w$ is perpendicular to $V$. (If you can't see why, draw a picture.) Now calculating the dot product of $(*{*})$ with $\v w_1$ and remembering that $\v w\cdot\v w_1=0$ gives you an equation for $\lambda_1,\lambda_2$. Calculating the dot product with $\v w_2$ gives another. Solve for $\lambda_1,\lambda_2$ and substitute back into $(*)$.

Good luck!