Say given the function:
$$ T: \mathbb R_{2}[t] \rightarrow \mathbb R_{1}[t]$$
$$ at^{2} + bt + c \rightarrow at + 2c – b $$
We were tasked to find the basis for the kernel of $T$ and $T(\mathbb R_2[t])$.
Where do I start? Could somebody point me in the right direction? I was told to do these steps:
But I don't even understand the first part. Could somebody explain how the matrix A is generated from writing the polynomials as column vectors? What is the map of T?
Best Answer
Let's see. You have to understand that $\mathbb{R}_2[t]$ is a vector space. A basis you could choose is $\{t^2, t, 1\}$ (it is easy to check that really it is). Analogously for $\mathbb{R}_1[t]$, a basis is $\{t, 1\}$. Now, in order to obtain the matrix of the transformation $T$, you only have to check that the first vector of the first basis ($t^2=1\cdot t^2+0t+0$) get transformed to $1.t+0$, so the first column of the matrix will be $(1,0)$ (as a column!). And so on for the others two element of the basis to create the other columns.