# Thread: Dimension of the subspace?

1. ## Dimension of the subspace?

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] }.

2. 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.

$\left|\begin{array}{ccc}
1&6&6\\
-1&-2&-1\\
0&3&4
\end{array}\right|=\ldots$

Good Luck

3. The dimension of a basis for a subspace, is just the number of vectors contained in the basis, so in your case this would be 3.

4. Originally Posted by skamoni
The dimension of a basis for a subspace, is just the number of vectors contained in the basis, so in your case this would be 3.
Be careful skamoni, where did the question ever say that the subset is a basis?
What if the set is linearly dependent?

5. Fair enough, row reduce it first then.

6. 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.