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Label the following statements as true or false. (a) The zero vector space has no basis

Chapter 1, Problem 1

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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

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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

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