Question

Find a basis for the span S of the vectors
(2, 2, 4, 3), (−2, 1, 2, 1), (3, 1, 2, 2), and (1, −1, −2, 0)

How would i do this?

Answer 1

Start by expressing the vectors as a matrix. Then use Gaussian Elimination to reduce the matrix to row-echelon form. Some of the columns will have no pivot, meaning they are dependent on other columns. Remove those, leaving only the linearly independant vectors. These form your basis.

Answer 2

How do i know which columns have no pivots?

Answer 3

@twvogels

Answer 4

When you reduce to row-echelon form, the pivots will show as '1's on the main diagonal. Any vector with no pivot will have a zero there instead.

Answer 5

so when i reduce it i get

1 0 0 -6
0 1 0 4
0 0 1 7
0 0 0 0

Answer 6

@twvogels so the last one has no pivot, so my basis is the first 3 vectors?

Answer 7

ie, 2, 2, 4, 3), (2, 1, 2, 1), (3, 1, 2, 2)