C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-3

# C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes4-3 -...

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SECTION 4.3 BASES FOR VECTOR SPACES Now, if we have a vector space V , we CANNOT talk about systems of equations, column vectors, matrices, or row operations. However, we CAN talk about the following for lists of vectors v 1 , . . . , v p . LINEAR INDEPENDENCE: The ONLY weights for which c 1 v 1 + c 2 v 2 + ··· + c p v p = 0 are c 1 = c 2 = ··· = c p = 0. LINEAR DEPENDENCE: There are weights, not all zero , for which c 1 v 1 + c 2 v 2 + ··· + c p v p = 0 . This equation is then called a linear dependence relation. BASIS (VERY IMPORTANT). A basis for a subspace H of a vector space V is an ordered list of vectors B = { b 1 , . . . , b p } which are linearly independent and span H . EXAMPLES. (1) The standard basis for R 5 (2) The standard basis for P 7 , the vector space of polynomials of degree no larger than 7. (3) The columns of an invertible n × n matrix A are a basis for R n .

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If we have a linear dependence relation, then we can solve for the last vector in the list with a nonzero weight (say
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## This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.

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