If y is an element of P3, then y is of the form:
ax^3 + bx^2 + cx + d
Notice there are four unknowns (a, b, c, d). This is the dimension of P3 (note: p3 is isomorphic to R4). The dimension of R2 is clearly 2. Since we are mapping a 4 dimensional space to a 2 dimensional space, the transform matrix is of dimension 2x4.
We know rank + nullity = # of columns.
The rank can be at most 2 here since there are only 2 rows (it can be 0, 1, 2 depending on tha matrix). Further, the matrix has 4 columns. Therefore, the nullity is at least 2 (2, 3, 4).
As a result, the nullity cannot be 0.