For each integer $j \ge 1$, let $V_j$ denote the real vector space of all polynomials in two variables of degree strictly less than $j$.What can be said about the dimension of the space of all linear maps from $V_5$ to $V_{11}$ whose kernel contains $V_3$ and whose image is contained in $V_7$?
I think, applying the rank nullity theorem, the rank of such a linear transformation should be less than or equal to $9$. But does rank equal the dimension of such linear maps? Does defining any sort of functional help here? Thanks beforehand.
Since the image is contained in $V_7$, that what you really want is the dimension of the space of all linear transformations from $V_5$ to $V_7$ whose kernel contains $V_3$. Besides, since $(1,x,y,x^2,xy,y^2,x^3,\ldots,y^4)$ is a basis of $V_5$ and since $(1,x,y,x^2,xy,y^2)$ is a basis of $V_3$, this is the same thing as the dimension of the space of all linear transformations from$$\operatorname{span}\bigl(\{x^3,x^2y,xy^2,y^3,x^4,x^3y,x^2y^2,xy^3,y^4\}\bigr)$$(which has dimension $9$) into $V_7$ (which has dimension $21$). Therefore, the answer is $21^9$.