I have spent nearly the last hour trying to do this and im really stuck, the question is:
Determine the dimension of the subspace spanned by the set
{ [1,−1,0], [6,−2,3], [6,−1,4] }.
Hi
You should check the value of the determinant of the three vectors. Either it is 0 and at least one vector is a linear combination of the two others. In this case ,the dimension of the subspace may be 0, 1 or 2 but has it is neither 0 nor 1 (why ?) it'll necessarily be 2. Either the value is not 0 and the subspace has dimension 3.
$\displaystyle \left|\begin{array}{ccc}
1&6&6\\
-1&-2&-1\\
0&3&4
\end{array}\right|=\ldots$
Good Luck
It's cheating a little bit, but I've found this program extremely useful for these questions:
Linear Algebra Toolkit
It's an on-line Linear Algebra programme, one of which determines whether set of vectors S is linearly independent or linearly dependent.
For your problem it found that
...the set S = {v1, v2, v3} is linearly independent.
Consequently, the set S forms a basis for span S.
which should help you determine the dimension of the subspace