Linear projection/R^3/Space for 3x-4y+6z is 0 and (2,0,5)/Exercise

Let ${}L\subseteq \mathbb {R} ^{3}$ be the solution space for the linear equation

${}3x+4y+6z=0\,,$

and set ${}W=\mathbb {R} {\begin{pmatrix}2\\0\\5\end{pmatrix}}$. Show that

${}\mathbb {R} ^{3}=L\oplus W\,$

holds, and describe the projections onto ${}L$ and onto ${}W$ with respect to the standard basis.

Create a solution