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 $v_1$, $v_2$, $v_3$."?
Best Answer
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.