Finding Range of a linear mapping

Question

In the following textbook example I understand how they get Null(L) but not how they get Range(L) which they say is clear to see. Can anyone elucidate the method of finding the Range of a linear mapping? Thank you!

Answer 1

The range of $L, R(L)\subseteq\Bbb P_3$. Every element in $R(L)$ is of the form $k_1x+k_2x^3$, which is a linear combination of $x$ and $x^3$. So $\{x,x^3\}$ is a basis of $R(L)$, giving its dimension as $2$.

Answer 2

You know $dim\ M(2,2)=4$ and $dim\ null(L)=2$, so by the Bank nullity theorem, $dim \ rank(L)=2$. But all the polynomials which are image of L are of the form $\alpha x + \beta x^3$, so you can get all of them with a linear combination of the polynomials $x$ and $x^3$. Since $dim \ rank(L)=2$ and $x,x^3$ are two linearly independent polynomials, it follows that $rank(L)$ is the subspace of $P_3$ generated by $x,x^3$.

Answer 3

$\{x,x^3\}$ is a linearly independent set and spans the range, as can be seen by looking at $cx+(a+b)x^3$. That's why it says "clearly".

In general the range of a linear map is the span of the columns of the matrix rel the standard basis, but we are able to take a short cut in this case.