Question

Let U be a vector space and V a subspace of U. (a) Assume dim(U) < ∞. Show that if dim(V ) = dim(U) then V = U. (b) Assume dim(U) = ∞ and dim(V ) = ∞. Give an example to show that it may happen that V 6= U.

Answer 1

Let U be a vector space and V a subspace of U.

(a). Let dim(V) = dim (U) = n, a finite number. and let {a₁,a₂,…,a_n} be a basis for V. Then the vectors a₁,a₂,…,a_n are linrearly independent in V and ,therefore, in U also, these vectors are linrearly independent . Further, since the dimension of U is n, hence the set {a₁,a₂,…,a_n} is a basis for U also. Therefore, every vector in U is a linear combination of the the vectors a₁,a₂,…,a_n so that U is a subset of V. However, since V is a subset of U, hence V = U.

(b). Let us consider the vector spaces U = (C,+) and V = (R,+) , i.e. the vector spaces of all complex numbers and all real numbers , under addition. Apparently, (R,+) is a subspace of (C,+). Further, both these vector spaces have infinite dimensions. However, (R,+) and (C,+) are not equal.