finding Ker(T) of a parameter's linear transformation

Question

I am suppose to find the ker(T) of linear transformation of: $$ G\begin{pmatrix}a & d \\ c & b\end{pmatrix}= a+\frac{b+c}{2}x+\frac{b-c}{2}x^2 $$

the form $T:V \to W$

My problem is that I don't really know how to read the right side of the equation, is that a vector? A polynomial?

Am I supposed to take the right side of the equation and make it equal with 0?

Answer 1

The transformation shown is a linear transformation from the space of $2 \times 2$ matrices to the space of polynomials of degree two-or-less, i.e., $V$ is the vector space of $2 \times 2$ matrices, and $W$ is the space of polynomials of degree two-or-less.

I echo Dietrich's comment: please post equations, not pictures. Search for "MathJax" to find out how to post equations in general.

Answer 2

Hint

$$G\begin{pmatrix}a & d \\ c & b\end{pmatrix}= a+\frac{b+c}{2}x+\frac{b-c}{2}x^2=0\iff a=b=c=0$$