Label the following statements as true or false.
(a) The zero vector space has no basis.
(b) Every vector space that is generated by a finite set has a basis.
(c) Every vector space has a finite basis.
(d) A vector space cannot have more than one basis.
(e) If a vector space has a finite basis, then the number of vectors in every basis is the same.
(f) The dimension of Pn(F) is n.
(g) The dimension of Mm×n(F) is m + n.
(h) Suppose that V is a finite-dimensional vector space, that S1 is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then S1 cannot contain more vectors than S2.
(i) If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way.
(j) Every subspace of a finite-dimensional space is finite-dimensional.
(k) If V is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimension n.
(l) If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent if and only if S spans V.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.