I've heard that for $\mathbb{R}^n$, there can only be $n$ number of basis vectors for it. Is that really the case though? Or am I having a misconception somewhere.
Let's take the example of $\mathbb{R}^4$. It seems that I can pick only 3 basis vectors $(1,1,0, 0)$, $(0,1,1,0)$ and $(0,0,1,1)$.
Firstly, these basis vectors are linearly independent. For example, no linear combination of $(0,1,1,0)$ and $(0,0,1,1)$ yields $(1,1,0,0)$. This is because these linear combinations will still have a zero as the first number in the vector.
Secondly, it (seems) that these 3 vectors span $\mathbb{R}^4$.