3
$\begingroup$

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...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.