Let D P2R P1R defined by Dp p Where p is the derivative o

Let D : P2(R) P1(R) defined by D(p) = p\' . Where p\' is the derivative of p. Find the null space of D. Is this map injective ?

Solution

Any constant scaling of that function will be zero as well.

Since the null space is not just 0, then any two vectors in the null space map to zero, thus it cannot be one one.

T(1,x,x²) = (0,1,x) The transformation doesn\'t contain the {0} therefore null D 0 therefore the map is not injective.