# Answer on Abstract Algebra Question for Sapna

Question #46089

a) Let V be the vector space of polynomials with real coefﬁcients and of degree at most 2.

If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,

ﬁnd [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.

b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators deﬁned by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.

i) Find T◦S and S◦T.

ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.

c) Find the inverse of the matrix 1 −1 0

2 −1 1

1 1 −1 using row reduction.

d) Let B1 ={(1,1),(1,2)}and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2.

If D = d/dx is the differential operator on V and B ={1+2x^2,x+x^2,x^2} is an ordered basis of V,

ﬁnd [D]B. Find the rank and nullity of D. Is D invertible? Justify your answer.

b) Let T: R^2 →R^2 and S: R^2 →R^2 be linear operators deﬁned by T (x1,x2) = (x1+x2, x1−x2) and S(x1, x2) = (x1, x1+2x2) respectively.

i) Find T◦S and S◦T.

ii) Let B ={(1,0),(0,1)}be the standard basis of R3. Verify that [T◦S]B = [T]B◦[S]B.

c) Find the inverse of the matrix 1 −1 0

2 −1 1

1 1 −1 using row reduction.

d) Let B1 ={(1,1),(1,2)}and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2.

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