Which the following are subspaces of P_2 with the usual operations? Span {1, x^2

Question

Which the following are subspaces of P_2 with the usual operations? Span {1, x^2} {a + ax: a elementof IR} {a + 1/b x: a, b elementof R, b notequalto 0} {ax^3: a elementof R} Select from the following: All of A, B, C and D. Only A, B, and D. Only A and B. Only B and D. None of the above. Which of the following sets are linearly independent? {(1, 0), (1, 1), (1, -1)} in R^2 {(1, 1, 1), (1, -1, 1), (-1, 1, 1)} in R^3 {1 + x, x, 2 + 3x} in P_2 {[1 1 0 0], [1 1 0 1]} in M_22 Select from the following: Only A and C. Only B. Only D. Only B and D. None of the above.

Explanation / Answer

Q No. 2.

Span { 1, x2} is obviously a subspace of P2 as { 1, x2} is the standardbasis for P2.

Span{a +ax: a R} is not a subspace of P2 as a+bx, when a b does not belong to set.

Span {a +x/b: a,b R} is not a subspace of P2 as the zero polynomial does not belong to this set.

Span { ax3 : a R} is not a subspace of P2 as ax3, when a 0, does not belong to P2 and hence this set is not a subset of P2.

Thus, the answer is only A is a subspace of P2. Option 5 is correct.

Q. No. 3.

A. This set is not linearly independent as (1,-1) = 2(1,0) -1(1,1)

B. This set is linearly independent as the RREF of the matrix with the given vectors as columns is I3

C. This set is not linearly independent as 2+3x= 2(1+x)+x.

D. This set is linearly independent as the 1st matrix is e1 +e2 while the 2nd matrix is e1 +e2 + e4

Thus only B and D are linearly independent. Option 4 is the correct answer.

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