Question

Define T : M2×2(R) → M2×2(R) by

T(A) = A − A^t .

a) Determine the rank and nullity of T.

b) Find bases for the nullspace N(T) and the range R(T) of T

c) Show that the only nonzero eigenvalue for T is λ = 2.

d) Find a basis for the eigenspace belonging to λ = 2.

Answer 1

Given that

(a) We know that is standard basis of M_{2,2}

For rank ,nullity of T ,first write standard matrix of T .

By (1),(2),(3),(4)

standard matrix of T is

Now for rank ,reduce above matrix to row reduced echlon form ,

INTERCHANGE R1 AND R2

R3->R3+R1

which shows that rank T is 1 ,by rank-nullity theorem

nullityT=4-1=3.

(b) Bases for nullT is given by since null space is the solution space of [T]x=0.

now choose x,y,w for basis

basis for kerT is

.

and basis for RangeT is (since dim(R(T)=1).

(c)

Suppose that is an eigen value of T ,then it will satisfy the characteristic equation of T,

that is only nonzero eigen value of T is 2 .

(d) For bases of eigen space corr to eigen value 2 ,suppose that u is an eigen vector corr to eigen value 2 ,then

Now choose u_3 for basis of eigen space

choose u_3=1

hence basis i s