I know this is a very simple question.
I am studying introduction to linear algebra by gilbert strang. But i cannot seem to understand this worked example.
Describe a subspace $S$ of each vector space $V$, and then a subspace $SS$ of $S$:
$V_1 =$ all combinations of $(1,1,0,0)$ and $(1,1,1,0)$ and $(1,1,1,1)$Answer:
$V_1$ starts with three vectors. A subspace comes from all combinations of the first two vectors $(1,1,0,0)$ and $(1,1,1,0)$. A subspace $SS$ of $S$ comes from all multiples $(c,c,0,0)$ of the first vector. So many possibilities.
2 questions:
Firstly, why is the subspace not $\mathbb{R}^4$(combination of all the vectors)? Since all would be able to pass through the origin.
What does it exactly mean by subspace of a subspace?
$\mathbb{R}^4$ can't be a subspace of $V_1$ because the vector $(1,0,0,0)$ is not in $V_1$ (i.e. it's not a linear combination of $(1,1,0,0)$, $(1,1,1,0)$ and $(1,1,1,1)$).
In general you can't span $\mathbb{R}^n$ with less than $n$ vectors.