Question

In: Math

Find the rank and the nullity of the linear transformation: T : P2 → P1, T(?...

Find the rank and the nullity of the linear transformation:

T : P2 → P1, T(? + ?? + ??^2) = (? + ?) + (? − ?)x

Solutions

Expert Solution

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.


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