Letbe a vector space over Q. Then which of the following (s) is/are correct for the vector space V(Q)?a)V is a finite dimensional vector spaceb)V is an infinite dimensional vector spacec)a subspace of V which has dimension 4d)a subspace of V which has dimension 2Correct answer is option 'B,C,D'. Can you explain this answer?

Question

Answer

Here in the elements of V components can be rational an irrational no. and field is Q.
So if each component is rational no. then corresponding to these elements spanning set of V has only 4 elements.
Since if

But when components are irrational no. then corresponding to these elements spanning set has infinitely many elements. Infact in this case each such element will be in the spanning set.

Let

then w₁ be a subspace of V. and dim w₁ = 4.
Let

Then w₂ also a subspace of V and dim w₂ = 2.

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