Find a basis for the subspace $\mathbb{R}^3$ containing vectors Let
$v_1 = \langle 1,0,-1\rangle$
$v_2 = \langle -2,7,2\rangle$
$v_3 = \langle 3,-7,-3\rangle$
I found that these are linearly dependent since I have a free variable upon reducing. However, the question asks to form a basis with those $3$ vectors. A basis can only be formed if all of the vectors are linearly independent.
How would I answer the following: "Find a basis for the subspace $\mathbb{R}^3$ containing $v_1$, $v_2$, $v_3$."?
 A: $v_3=v_1-v_2$ but $\{v_1,v_2\}$ are independ so they can be a basis for this subspace (that contain this three vectore)
A: I think you're overthinking this.  These three vectors lie in a common plane.  A plane requires only two basis vectors to be completely described.  Therefore, you can pick any two vectors that are linear combinations of the given set of 3, and you're done.
In other words, choose two basis vectors $b_1, b_2$ and express them as linear combinations of $v_1, v_2, v_3$.  You have freedom to choose what $b_1, b_2$ are, provided that they are linearly independent of one another.
A: I'm just a student and get confused on the grammar but my comment is this: 
For a basis for R^3 you need 3 L.I. vectors to span R^3 not two. To get the third you can do the dot product with the first two to get a third L.I. vector letting one of the variables being anything. The point is this 3rd vector will be L.I. from the first two and not lie in the plane of the first two and therefore span R^3.
