Bases of subspaces

Question

I need some help with this homework question.

How do I determine whether the set of $\{(1,-1,0),(0,1,-1)\}$ forms a basis for the subspace of $\mathbb{R}^3$ consisting of all $(x,y,z)$ such that $x+y+z=0$?

I know that for a vector space to form a basis, it has to span $V$ and be linearly independent.

Do correct me if I'm wrong in any areas

Thanks in advance :)

Answer 1

Check if your two vectors are linearly independent (that is if they are not collinear). Then check that both vectors belong to your subset. If you can answer positively to both questions then you are done since your subspace has dimension 2 (therefore any linearly independent collection of vectors of $V$ of exactly two vectors is a basis).

Answer 2

Here a hint: a generic vector in the subspace you consider is $$(x,-x-z,z) = x(1,-1,0)+z(0,-1,1).$$

Answer 3

This set subspace is the space of all vectors of the form,

$$v = \begin{pmatrix}a\\ b\\ -a-b\end{pmatrix}$$

which is obviously a 2 dimensional space. Now, we need to be able to write $v$ as a linear combination of our proposed basis vectors.

Answer 4

If you answer yes to the following, then the full answer is yes! Let $V$ be the vector space defined by $V = \{(x,y,z) \mid x+y+z=0\}$

1) Are the vectors you are checking as a basis linearly independent?

2) Are the vectors you are checking as a basis in $V$?

3) If (1) and (2) are both yes, then the final question to answer is: is the dimension of $V$ equal to the number of vectors you are checking as a basis?