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Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1.$

(a) Prove that the matrix $$\mathbf{A} = \begin{pmatrix} -2 & -2 & 2 \\ 2 & 0 & 1 \\ 0& -1 & -1 \end{pmatrix} $$maps all points on line $\ell$ to points on plane $P$.

(b) Prove that the matrix $$\mathbf{B} = \begin{pmatrix} 1 & 1 & -1 \\ 3 & 3 & -1 \\ 5& 5 & -1 \end{pmatrix}$$maps all points on plane $P$ to points on line $\ell$.

I've figured out part A but need help with part B. I know this question has been asked before at Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1$., but I have a few questions about the answer there.

  1. Why must we multiply matrix B by $\begin{pmatrix} x \\ y \\ 1-x-y \end{pmatrix}$?
  2. Assuming we do multiply matrix B and $\begin{pmatrix} x \\ y \\ 1-x-y \end{pmatrix}$, the result we get is $\begin{pmatrix} 2x + 2y - 1 \\ 4x+4y - 1 \\ 6x+6y-1 \end{pmatrix}.$ I don't think that lies on $(t, 2t+1, 3t+2)$....
StubbornAtom
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riorpewruiowe
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1 Answers1

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You can describe a point on a plane by two independent coordinates. They choose those coordinates to be $x$ and $y$. Note that $z$ in this case is tied to the first two by the equation of the plane. So a point $R(x,y)$ in this plane is described by $$R(x,y)=(x,y,1-x-y)$$ Now all you need to do is transform this point using the $B$ matrix to a point on the line.

For your second question, just choose $$t=2x+2y-1$$Then $$4x+4y-1=2(2x+2y-1)+1=2t+1$$ Similarly, you can get the last coordinate. Note that an entire line in the plane is mapped to the same point on the line.

Andrei
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