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MAT1341-L07-Bases - BASES JOSE MALAGON-LOPEZ Given a vector...

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BASES JOS ´ E MALAG ´ ON-L ´ OPEZ Given a vector space V = Span( S ), the generator set S might not be good enough when we want to work with V . One problem is that a vector v in V might be described in more than one way in terms of the generator set. For example, consider the subspace Span( S ) of R 3 , where S = 1 2 - 1 , 2 - 3 - 5 , - 1 5 4 The vector 1 9 2 is in Span( S ) and can be expressed as a linear com- bination of the vectors in S in different ways: 1 9 2 = 1 1 2 - 1 + 1 2 - 3 - 5 + 2 - 1 5 4 = 3 1 2 - 1 - 1 2 - 3 - 5 + 0 - 1 5 4 = 2 1 2 - 1 + 0 2 - 3 - 5 + 1 - 1 5 4 Another problem is that we might be working with more vectors than what is required: if v is a linear combination of the vectors in { v 1 , . . ., v m } , then Span ( v 1 , . . . , v m ) = Span ( v 1 , . . . , v m , v ). 1
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For example, since 1 2 - 1 - 2 - 3 - 5 = - 1 5 4 we have that Span 1 2 - 1 , 2 - 3 - 5 , - 1 5 4 = Span 1 2 - 1 , 2 - 3 - 5 The good news is that a vector space is completely determined when we choose a generator set that is linearly independent as well. Bases Definition. A basis for a vector space V is a linearly independent subset of V that generates V . Examples. The empty set is a basis for { 0 } . A basis for R n is { ~ e 1 , . . . ,~ e n } , called the standard basis. A basis for P n is { 1 , x, x 2 , . . . , x n } , called the standard basis. A basis for M m × n is the set of matrices { A kl = ( a ij ) | a ij = 1 if i = k, j = l, and a ij = 0 otherwise } , called the standard basis. For instance, a basis for M 2 × 3 is 1 0 0 0 0 0 , 0 1 0 0 0 0 , 0 0 1 0 0 0 , 0 0 0 1 0 0 , 0 0 0 0 1 0 , 0 0 0 0 0 1 Any linearly independent set S is a basis for Span( S ). A basis for P is { 1 , x, x 2 , . . . , x n , . . . } .
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