Question 106518
Remember to find a basis, we need to find which vectors are linear independent. So take the set *[Tex \LARGE \left\{\left[ 1 \\ -1 \\ 0\right],\left[ -1 \\ 0 \\ 1\right] \left[ 0 \\ 1 \\ -1\right]\right\}]


and form the matrix


{{{(matrix(3,3,1,-1,0,-1,0,1,0,1,-1))}}}


Now use Gaussian Elimination to row reduce the matrix 


{{{(matrix(3,3,1,-1,0,0,1,-1,-1,0,1))}}} Swap rows 2 and 3



{{{(matrix(3,3,1,-1,0,0,1,-1,0,-1,1))}}} Replace row 3 with the sum of rows 1 and 3 (ie add rows 1 and 3)



{{{(matrix(3,3,1,-1,0,0,1,-1,0,0,0))}}} Replace row 3 with the sum of rows 2 and 3 (ie add rows 2 and 3)



{{{(matrix(3,3,1,0,-1,0,1,-1,0,0,0))}}} 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

*[Tex \LARGE \left\{\left[ 1 \\ -1 \\ 0\right],\left[ -1 \\ 0 \\ 1\right]\right\}]

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


So the basis is:


*[Tex \LARGE \beta=\left\{\left[ 1 \\ -1 \\ 0\right],\left[ -1 \\ 0 \\ 1\right]\right\}]