SOLUTION: The questions is Find a basis for the span of the given vectors. [1, -1, 0], [-1, 0, 1], [0, 1, -1]. For some reason I reduced it and got [1 0 0], [0 1 0], [-1 -1 0] but i

Algebra ->  Matrices-and-determiminant -> SOLUTION: The questions is Find a basis for the span of the given vectors. [1, -1, 0], [-1, 0, 1], [0, 1, -1]. For some reason I reduced it and got [1 0 0], [0 1 0], [-1 -1 0] but i      Log On


   

Question 106518: The questions is
Find a basis for the span of the given vectors.
[1, -1, 0], [-1, 0, 1], [0, 1, -1].
For some reason I reduced it and got
[1 0 0], [0 1 0], [-1 -1 0]
but i am not sure where/how to go from here and the book that we have does't give an example of how to do this. Was I even supposed to reduce it?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember to find a basis, we need to find which vectors are linear independent. So take the set

and form the matrix

%28matrix%283%2C3%2C1%2C-1%2C0%2C-1%2C0%2C1%2C0%2C1%2C-1%29%29

Now use Gaussian Elimination to row reduce the matrix

%28matrix%283%2C3%2C1%2C-1%2C0%2C0%2C1%2C-1%2C-1%2C0%2C1%29%29 Swap rows 2 and 3


%28matrix%283%2C3%2C1%2C-1%2C0%2C0%2C1%2C-1%2C0%2C-1%2C1%29%29 Replace row 3 with the sum of rows 1 and 3 (ie add rows 1 and 3)


%28matrix%283%2C3%2C1%2C-1%2C0%2C0%2C1%2C-1%2C0%2C0%2C0%29%29 Replace row 3 with the sum of rows 2 and 3 (ie add rows 2 and 3)


%28matrix%283%2C3%2C1%2C0%2C-1%2C0%2C1%2C-1%2C0%2C0%2C0%29%29 Replace row 1 with the sum of rows 1 and 2 (ie add rows 1 and 2)


Now the matrix in reduced row echelon form. Notice the matrix only has 2 pivot columns (which are the first two columns). This means the first two columns of the original matrix are linearly independent. Since the third column does not have a pivot, it is dependent on the first two columns


So to form a basis, simply pull out the linearly independent columns of the original set of vectors to get the set

this set will span the original set (since taking out a dependent vector does not change the span). Also since the set is linearly independent, this set forms a basis (since both properties are satisfied)

So the basis is: