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October 31, 2014

Homework Help: linear algebra

Posted by chris on Thursday, July 19, 2007 at 2:02pm.

which of the following sets of vectors span R^3?
a.){(1, -1, 2), (0, 1, 1)}
b.) {1, 2, -1), (6, ,3, 0), (4, -1, 2), (2, -5, 4)}
c.) {(2, 2, 3), (-1, -2, 1), (0, 1, 0)}
d.) {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)}

can someone show the steps to check for one of them and i will try to do the rest.

Put the vectors as rows or columns in a matrix and do Gaussian Elimination (note that row rank = column rank).

You don't have to do this for a) because you can't span R^3 with two vectors. Also, it is clear that the vectors listed in d) span R^3

In case of b) after Gaussian Elimination, you should find that the rank of the matrix is 2. This means that te four vectors span a two dimensional subspace of R^3 (the reduced matrix indicates exactly what subpace)

In case of c) you should find that the rank is 3, so the three listed vectors span R^3.

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