Given $(1,2,3)$, $(4,5,6)$, and $(7,8,9)$.
Which is independent?
A. $(0,0,0)$ B.$(1,0,1)$ C.$(1,1,1)$ D. $(2,4,6)$ E.$(3,5,7)$
Clearly the set of given vectors is not linearly independent. My question is if they still span a subspace of $\Re^3$. If so, does this help in determining which vector will be independent?
On
It seems that you already know that the given vectors span a $2$-dimensional subspace of $\mathbb{R}^3$, so we are interested in the linear systems
$$x+4y=a\\2x+5y=b\\3x+6y=c$$
where $(a,b,c)$ is one of the answer choices. The reduced form of the augmented coefficient matrix is
$$\begin{pmatrix}1&0&\frac{4b-5a}{3}\\0&1&\frac{2a-b}{3}\\0&0&-a+2b-c\frac{}{}\end{pmatrix}$$
Plugging in different values for $(a,b,c)$ we find that only $(1,0,1)$ makes the system inconsistent.
HINT: Every subset of $\Bbb R^3$ spans a subspace of $\Bbb R^3$. In this case you could show that $$\{(1,2,3),(1,1,1)\}$$ is a basis for the subspace spanned by the original three vectors and then easily show that all but one of the choices are linear combinations of $(1,2,3)$ and $(1,1,1)$.