Let $V=P_2(\mathbb{R})$ be a vector space over $\mathbb{R}$.
Let $B={1,t,t^2}$ be the standard basis of $V$ and consider the ordered set $B1={1+t,t+t^2,1+t^2}$ of $V$.
Question 1) Prove $B1$ is a basis of $V$.
Question 2) Determine the transition matrix from $B$ to $B1$ and $B1$ to $B$.
Question 3) Let $v=1+2t+3t^2$. Determine the coordinate vectors $[v]_B$ and $[v]_{B1}$.
I have done question 2 but really don't know where to start with question 1.
For question 3 I have written $v$ in terms of $B_1$ to get $[v]_{B1}$ but not sure how to work out $[v]_B$ from this? I'm not sure if I'm missing a trick or something!
You need to check two things:
If the answer to the first question is 'yes', then you could skip the second one if you know that the dimension of $P_2\left( \mathbb{R} \right)$ is equal to 3; and if you are allowed to use this of course. If not, there is still the manual approach and this will involve solving a linear system (3 by 3).
The 'manual approach' would be to write $1+2t+3t^2$ as a linear combination of the elements of $B_1$ (use unknown coefficients) and solve this system of three linear equations in the three unknown coefficients.
But since you set up the transition matrices in question 2, you can of course use this result!
Addition after comment.
To check whether $1+t$, $t+t^2$ and $1+t^2$ are linearly independent, you look at the following system of equations: $$a \left( 1+t \right) + b \left(t+t^2 \right) + c \left( 1+t^2 \right) = 0 \iff \left\{ \begin{array}{rcl} a+c & = & 0 \\ a+b & = & 0 \\ b+c & = & 0 \end{array}\right.$$ They are linearly independent if the only solution to this system is the trivial zero solution (so $a=b=c=0$).