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 inevery basis is the same.(f) The dimension of Pn(F) is n.(g) The dimension of Mmxn(F) ism+ n .(h ) Suppose that V is a finite-dimensional vector space, that S1 is alinearly independent subset of v, and that s2 is a subset of v thatgenerates V. Then S1 cannot contain more vectors than S2 . (i) If S generates the vector space V, then every vector in V can bewritten as a linear combination of vectors in S in only one way.(j) Every sub pace of a finite-dimensional space is finite-dimensional.(k ) If Vis a vector space having dimension n, then V bas exactly one subspace with dimension 0 and exactly one subsp ace with dimension n . (1) If V is a vector space having dimension n, and if S is a subset ofV with n vectors, t hen S is linearly independent if and only if Sspans V.
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
every basis is the same.
(f) The dimension of Pn(F) is n.
(g) The dimension of Mmxn(F) ism+ 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 sub pace of a finite-dimensional space is finite-dimensional.
(k ) If Vis a vector space having dimension n, then V bas exactly one subspace with dimension 0 and exactly one subsp ace with dimension n .
(1) If V is a vector space having dimension n, and if S is a subset of
V with n vectors, t hen S is linearly independent if and only if S
spans V.
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