In: Math
Find the rank and the nullity of the linear transformation:
T : P2 → P1, T(? + ?? + ??^2) = (? + ?) + (? − ?)x
T: P2 → P1 is defined by T(?+ bx + ?x2) = (? + ?) + (? − ?)x.
sets S2 = {1,x,x2} and S1 = {1,x} are the standard bases for P2 and P1 respectively. Also, T(1) = 1, T(x) = 1+x and T(x2) = -x. Hence the standard matrix of T is A (say) =
1 |
1 |
0 |
0 |
1 |
-1 |
It may be observed that the entries in the colu7mns of A are the scalar multiples of 1 and the coefficients of x respectively.
The RREF of A is
1 |
0 |
1 |
0 |
1 |
-1 |
It implies that the first 2 columns of A are linearly independent and the 3rd column of A is a linear combination of the first 2 columns .
Therefore, the rank of T = rank(A) = 2 and the nullity of T = nullity of A = no. of columns in A -rank(A) = 3-2 = 1.