Let V be a 3 dimensional vector space with A and B its subspaces of dimensions 2 and 1 , respectively. If, thena)V = A -Bb)V = A +Bc)dim(A + B) ≤ 3d)dim( A ∩ B) ≠ 0Correct answer is option 'B'. Can you explain this answer?

Question

Let V be a 3 dimensional vector space with A and B its subspaces of dimensions 2 and 1 , respectively. If, thena)V = A -Bb)V = A +Bc)dim(A + B) &le; 3d)dim( A &cap; B) &ne; 0Correct answer is option 'B'. Can you explain this answer?

Answer

We know

dim A + dim B = dim (A + B) + dim (A ∩ B ) .............. (1)

Here A ∩) B = { 0 } => dim (A ∩ B ) = 0

then by equation (1), dim (A + B ) = 3 => option (C) and (D) are incorrect.

Now (A + B) be a subspace of V and has same dimension as V.

=> V = A + B

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