Show that given vectors are not vector space basis in $R^4$

Question

The task is to show that the vectors $(1, 0, 0, 0)$, $(1, 1, 0, 0)$ and $(1, 1, 1, 0)$ are not vector space basis in $R^4$. I tried the method of getting a system of linear equations: $\alpha(1, 0, 0, 0) + \beta(1, 1, 0, 0) + \gamma(1, 1, 1, 0) = (0, 0, 0, 0)$, which gives the result $\gamma = 0$ and consequently $\alpha = 0$ and $\beta = 0$ - and if I understand correctly, this would mean that the given vectors are the vector space basis in $\mathbb R^4$. How do I prove these vectors are not the vector space basis?

Answer 1

With your approach, what you can prove is that they are linearly independent — and they are. But in order be be a basis, they must span $\mathbb{R}^4$, and they don't. The space $\mathbb{R}^4$ has dimension $4$ and therefore no set with less than $4$ elements can span it.

Answer 2

You need to show that your set is spanning:

Consider the vector $(0,0,0,1)$. Is it a linear combination of those vectors? This is a consequence of the fact the the dimension of $\mathbb{R}^4$ is $4$, so you need at least $4$ vectors to span it, as @Jose Carlos Santos pointed out.