basis and dimensions.pdf

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Math211 M.Sabat Basis (pl: bases) and dimension. Definition: The vectors v 1 ,…..,v n in a vector space V are said to form a basis for V, if: a) these vectors span V b) these vectors are linearly independent in V. A vector space may have more then one basis, but all bases will contain the same number of elements. This number is said the dimension of the vector space. Notation: dim V. Dim({0})=0 Knowing a spanning set for a vector space, we can define two kinds of bases: a) basis that is subset of the spanning set . b) Basis that is not subset of the spanning set . We can also find a basis that is not subset of the spanning set such that none of its vectors belong to the spanning set. To find a basis for a given vector space, we should first find a spanning set , and then find linearly independent vectors between these vectors (for bases that are subsets of the spanning set) or use the spanning vectors to construct another set of vectors that also span the vector space and that is linearly independent (for bases that are not subsets of the spanning set).
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