notes-bases

notes-bases - Math 1b Prac Bases for, and dimensions of,...

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Math 1b Prac — Bases for, and dimensions of, subspaces of R m January 23, 2009 It is fundamental in linear algebra that all bases of a subspace have the same number of vectors. In these notes, we establish this for subspaces of R m . This can be seen quickly using row operations. For subspaces of abstract vector spaces, some additional ideas are required. This additional material is done carefully in the text and we will do some of this in class later. Recall that row operations on a matrix M do not change the row space or the nullspace (solution space) of M . Th isisimportant . I like to say the “column relations” are not changed by row operations. For example, if the 5th column of M is the sum of the 2nd column and 5 times the 3rd column, then the same relation will be valid for any matrix M 0 that is row-equivalent to M .T h i si s because the stated relation among the columns of M is equivalent to the statement that (0 , 1 , 5 , 0 , 1 , 0 , 0 ,... ) is in the nullspace of M . This vector will be in the nullspace of any row-equivalent matrix M 0 . To decide whether vectors v 1 , v 2 ,....
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notes-bases - Math 1b Prac Bases for, and dimensions of,...

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