Determine whether the given set S is a subspace of the vector space V. A. V = M_

Question

Determine whether the given set S is a subspace of the vector space V. A. V = M_3 (R) and S is the set of 3 times 3 matrices with trace 0 (recall the trace of a matrix is the sum of its diagonal entries) B. V = M_3(R) and S is the set of 3 times 3 matrices of rank 1 C. V = M_3 (R) and S is the set of 3 times 3 matrices A such that the vector (7 4 3) is in the of A D. V = M_2(R) and S is the set of 2 times 2 matrices A such that A^2 = O (here O is the 2 times 2 zero matrix) E. V = M_2 (R) and S is the set of 2 times 2 matrices that commute with the matrix (0 5 1 6) F. V = M_3(R) and S is the set of 3 times 3 matrices A such that A^2 = A

Explanation / Answer

A) Let A, B in S

Then tr(A) = 0 and tra(B) = 0

tra(A+B) = tra(A)+ tr(B) = 0

Also for any scalar k , tra(kA) = ktra(A) = 0

Hence A+B and kA is in S for A,B in S. So S is a subspace of V

B) Let A, B in S

Then rank(A) = 1 and rank(B) =1

Now rank(A+B) < = rank(A) + rank(B) = 1+1 = 2

Hence rank(A+B) is either 1 or 2 and so A+B need not be in S

S is not subspace of V

C) Let A, B in S

Then (7 4 3)T is in the nullspace of A and B

Then (7 4 3)T is also in nukk space of A+B

Also as any vector k(7 4 3)T is in nullspace and hence kA is in S

So S is subspace of V

D) Consider , A, B in S

Then A2 = 0 and B2 = 0

(A+B)2 = A2 + B2+ AB + BA = 0+0+AB + BA = AB + BA

Hence (A+B) is in S only if AB also equals BA equals 0

So S is not a subspace of V

E) Let A, B in S

Then A and B commute with given matrix T(say)

Now (A+B)(T) = AT + BT = TA + TB = T(A+B)

Hence A+B also commute with given matrix T

Consider (kA)T = k(AT) = k(TA) = (kTA) = T(kA)

Hence kA also commute with given matrix T

So S is subspace of V

F) Let A, B in S

A2 = A, B2 = B

(A+B)2 = A2 + B2 + AB + BA = A+B + AB + BA

Hence S is not a subspace of V

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