Let the following be the linear transformation for $T$ from a $2$ x $2$ matrix to a second degree polynomial.
$T\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\left(a-c\right)x^2+c\left(x-1\right)+b$
Find a basis for $Im(T)$ and $Ker(T)$.
So this is what I did to find the kernel.
I rearranged the equation to get:
$\left(a-c\right)x^2+\left(c\right)x+\left(-c+b\right)$
So I tried to set everything in the brackets to $0$ and then got that:
$a=c=0,\:b=0,\:d=free$
This led me to think that a basis for $Ker(T)$ would be:
$\left\{\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\right\}$
For the $Im(T)$, I rearranged the equation to get:
$a\left(x^2\right)+b+c\left(-x^2+x-1\right)$
This gave me the following basis:
$\left\{x^2,\:1,\:-x^2+x-1\right\}$
Which had the following matrix:
$\begin{pmatrix}1&0&-1\\ 0&0&1\\ 0&1&-1\end{pmatrix}$
But all three columns of this matrix are linearly independent, which doesn't make any sense as the kernel I have calculated has dimension $1$.
Does anyone know where I have gone wrong here?
Thank you!
Best Answer
When you say, "Which has the following matrix", have a think about what you mean by this. You've taken a basis of polynomials (not a linear transformation) and somehow turned it into a matrix of numbers. What does this matrix mean? Why do you expect the columns to be linearly independent?
What you have done is form the change of basis matrix from the given basis to the standard basis, in the order $(x^2, x, 1)$. Such a matrix must certainly be invertible; change of basis matrices always are, and their inverse is the change of basis matrix from the second basis to the first.
Your issue seems to suggest that you're looking to find a matrix for the transformation, between the two bases. Bear in mind that this is impossible, since the basis is for the kernel and range, both of which are respectively subspaces of the domain and codomain, and the kernel happens to be a strict subspace. You would need full bases for the domain and codomain before you can compute a matrix for the transformation.
But, this isn't what the question wanted. It asked for two bases for the kernel and range, and you have found exactly that.