Linear mapping and linear span

Question

This is about a linear algebra problem.
We have $f\in L(\mathbb{R^3},\mathbb{R^2})$ such that :

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

We need to find $f(x,y,z)$.

My professor started with $(x,y,z)=\alpha(1,0,0)+\beta(1,1,0)+\gamma(1,1,1)$ but isn't this supposing that $f$ is mapping from $<F>$ not $\mathbb{R^3}$ (with $F=\{(1,0,0), (1,1,0), (1,1,1)\}$)?

Answer 1

Hint: Find $f(0,1,0)$ and $f(0,0,1)$ by linearity of $f$.