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..

1. 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.