Find the transformation.

Question

I have to define (find?) the linear transformation $ f:\mathbb{R}^{3}\rightarrow \mathbb{R}^{2} \ \ \ where:$

$f(1,1,0)=(1,1)$

$f(0,2,-1)=(-1,0)$

$f(1,2,-1)=(0,2)$

How to achieve this? It is hard to guess this transformation I think. I will be glad if someone give me few tips or method how to do this task.

Answer 1

The transformation $f$ has this form $$f(x,y,z)=(a_1x+b_1x+c_1z,a_2x+b_2x+c_2z)\tag{*}$$ write the given equalities using $(*)$ and solve for the coefficients $a_i,b_i$ and $c_i$.

Answer 2

Hint: Note that $f$ will be of the form $f(x) = Ax$ where $A$ is a $2 \times 3$ matrix, say $$A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix}$$ Use this fact to rewrite the three givens into a system of linear equations.