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 3^rd 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.

milcah answered 1 year ago

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