Linear Algebra Answers

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If a linear operator is diagonalisable, its minimal polynomial is the same as the
characteristic polynomial.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If zero is an eigenvalue of a linear transformation T, then T is not invertible.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If the characteristic polynomial of a linear transformation is (x-1)(x-2), its
minimal polynomial is x-1 or x-2.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

The row-reduced echelon form of an invertible matrix is the identity matrix.

Which of the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

The rank of a matrix equals its number of nonzero rows.

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample

If W1 and W2 are subspaces of vector space V and W1+W2 =V, thenW1∩W2 = {0}

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If {v1, v2, .... , vn} is a basis for vector space V, then {v1+v2+ +vn, v2,.... ,vn} is also
a basis for V

Q.Give some detail explanation on Pseudo inverse matrix???

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 = f(1, 0), (0, 1) be the standard basis of R^3. Verify that
[T ◦ S]B = [T]B ◦ [S]B.

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.