For clarity this is a coursework question so I'm not looking for an answer and more of a nudge in the general direction.
The question starts
The map $\theta :\Bbb{R}_3 [t] \rightarrow \Bbb{R}^3\ $ is given by $at^3+bt^2+ct+d\mapsto \begin{bmatrix}2a-4c\\0\\6c-3a\end{bmatrix}$
The first part of the question involve finding a basis for the kernel of $\theta$ which I found to be the set $B = \left\{(1),(t^2),(t^3+t/2)\right\}$
The next part then asks me to find a basis of $\Bbb{R}_3[t]$ that contains $B$
I am completely stuck as to what to do here and would appreciate any help, thanks.
Best Answer
Hint
Theorem: any linearly independent family of vectors can be completed into a basis by picking up vectors from an existing basis.
So take a basis of $V=\mathbb R_3[t]$... there is a famous one! And pick up one of the vector of this famous basis to transform the linearly independent family $B$ of vectors that generates $\ker \theta$ into a basis of $V$.
Additional hint... in your pick up process, some vectors are better than the others.