Find the matrix of the linear transformation $T : \Bbb R^3 \to \Bbb R^3$ that projects a vector onto the plane $x − y + 2z = 0$

Question

Find the matrix of the linear transformation $T : \Bbb R^3 \to \Bbb R^3$ that projects a vector onto the plane $x − y + 2z = 0$.

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I know how to find matrix of linear transformation when projecting onto the xy/xz/yz planes but not sure what to do when then plane is an equation. Any help would be greatly appreciated.

Answer 1

Hint:

An orthonormal basis for your plane $x-y+2z=0$ is given by $$v_1 = \frac1{\sqrt2}(1,1,0), \quad v_2 = \frac1{\sqrt3}(1,-1,1)$$ so your $T$ is given by $Tx = \langle x,v_1\rangle v_1 + \langle x,v_2\rangle v_2.$

Answer 2

Here is an outline of an approach slightly ddifferent from mechanoidroid's.

First, note that the vector $v=(1,-1,2)$. is orthogonal to the plane $x-y+2z=0$ (why?). Next construct a map $P$ that projects onto the line spanned by $v$. Finally, note that if $T$ is the projection onto the plane, $P+T=I$ (why?), similar to how projection onto the z axis and projection onto the xy-plane add to $I$.