Suppose B is a basis for a real vector space V of dimensiongreater than 1. Which

Question

Suppose B is a basis for a real vector space V of dimensiongreater than 1. Which of the following statements could betrue. a.) Then zero vector of V is an element of B b.) B has a proper subset that spans V c.) B is a proper subset of a linearly independent subset ofV d.) There is a basis for V that is disjoint from B e.) One of the vectors in B is alinear combination of theother vectors in B Suppose B is a basis for a real vector space V of dimensiongreater than 1. Which of the following statements could betrue. a.) Then zero vector of V is an element of B b.) B has a proper subset that spans V c.) B is a proper subset of a linearly independent subset ofV d.) There is a basis for V that is disjoint from B e.) One of the vectors in B is alinear combination of theother vectors in B

Explanation / Answer

(a) zero vector can never be a member of the basis while thevectors in the basis are linearly independent and zero vector isdependent. (b) while B is the basis of V, any proper subset of B cannotspan V. (c) if V is a vector space of dimension n, then any subset ofV with n vectors can be linearly independent and a subset with n+1or more vectors is dependent. further, dimension of the vector space is nothing butthe number of vectors in the basis. basing on these facts, we can say that the maximum numberof linearly independent vectors possible in a set is nothingbut a basis. in other words, B is the basis and so B has maximum number oflinearly independent vectors in a subset of V. (d) there can be infinite basis to V other than B. but any two basis have equal number of vectors. (e) while the vectors in B are linearly independent, no vectorin B can be a linear combination of the other in the sameset.

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