申論題

8. (10%) How many of the following five statements are correct? ___________
ㆍLet V be a vector space. Let S be a subset of V. Let U be a subspace of V. IfS U,

then span(S) U.

ㆍIf R is a linearly dependent subset of a vector space, then x e span(R | [x])) holds

for each vector x R.

ㆍBased on any consistent axiom set for set theory, any vector space admits a basis.

ㆍBased on the standard ZFC axioms for set theory, each inner-product space admits

an orthonormal basis.

ㆍFor any complex vector spaces V and W with dim(W) <∞ , if T is a linear surjec-

tion from V to W, then dim(V)= nullity(T) + rank(T).