# Linear Algebra basis

Let $$f\in(\Bbb R^3)^*$$ be defined by $$f(x_1,x_2,x_3)=x_1+2x_2-x_3$$ and $$g:\Bbb R^2\mapsto \Bbb R^3$$ $$(x_1,x_2)\mapsto (3x_1-2x_2,x_2,x_1-2x_2)$$

1. Determine the matrix $$M_{B_2^*\leftarrow B_3^*}(g^*)$$ where $$B_2^*$$ and $$B_3^*$$ are the standard basis in $$\Bbb R^2$$ and $$\Bbb R^3$$

2. Determine $$g^*(f)$$

So for a), if g takes something in $$\Bbb R^2$$ and outputs $$\Bbb R^3$$, then $$g^*$$ would take somehting from $$\Bbb R^2$$ to $$\Bbb R$$ or from $$\Bbb R^3$$ to $$\Bbb R$$? I am not really sure how the lienar functional would look like! Any help appreciated.

EDIT:

If g is a linear function, I.e. $$g:\Bbb R^2 \rightarrow \Bbb R^3$$, how can it have a dual space ? It’s not a vector space itself...