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Label the following statements as true or false. (a) The zero vector space has no basis
Chapter 1, Problem 1(choose chapter or problem)
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. 54 Chap. 1 Vector Spaces (e) (f) (g) (h) (i) (j) (k) (1) If a vector space has a finite basis, then the number of vectors in every basis is the same. The dimension of Pn (F) is n. The dimension of MmXn (F) is rn 4- n. Suppose that V is a finite-dimensional vector space, that Si is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then Si cannot contain more vectors than S2. 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. Every subspace of a finite-dimensional space is finite-dimensional. 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. 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
Questions & Answers
QUESTION:
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. 54 Chap. 1 Vector Spaces (e) (f) (g) (h) (i) (j) (k) (1) If a vector space has a finite basis, then the number of vectors in every basis is the same. The dimension of Pn (F) is n. The dimension of MmXn (F) is rn 4- n. Suppose that V is a finite-dimensional vector space, that Si is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then Si cannot contain more vectors than S2. 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. Every subspace of a finite-dimensional space is finite-dimensional. 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. 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
ANSWER:Step 1 of 2
a) False
b) True
c) False
d) False
e) True
f) False
g) False