I have just started to learn QC. It is said that
The quantum state of $N$ qubits can be expressed as a vector in a space of dimension $2^N$
If there is $1$ qubit then we have two possible state vectors $|0\rangle$ and $|1\rangle$ or $(0,1)$ and $(1,0)$ respectively. Getting to $2$ qubits we have $4$ possible state vectors $(1,0,0,0), (0,1,0,0), (0,0,1,0), $ and $(0,0,0,1)$. Note that in each case, all entries are zero except 1. The point I am trying to get to is that:
$2^N$ seems like a big space but given a vector in this space - all components will be zero except $1$. So there are only $2^N$ possible values the state vector can take. Is this not correct? If not, why?
Why don't we say the space is $N$-dimensional. A $N$-bit string has $2^N$ possible values.