Maximum number of pairwise linearly independent vectors

Question

Consider vectors $v_1,\dots,v_n\in\mathbb{R}^d$. My question is: What is the maximum number of such vectors, that are pairwise linearly independent?

Clearly, if we remove the word pairwise the answer is $d$, but it feels the number is larger. Is this known explicitly?

Answer 1

Consider the case $d=2$. Then you have that the vectors $(1,0)$, $(0,1)$ and $(\sqrt{2},\sqrt{2})$ are pairwise linearly independent vectors, for example. Indeed, the set of all vectors of length $1$ (unit circle) consists of pairwise linearly independent vectors (except, of course, those pairs on the same line). Notice that such set is uncountable.

The same argument can be extended to any dimension $d$. Hence, the maximum number is infinity.

Answer 2

In dimension $2$ the uncountable set of all vectors $(x,y)$ of length $1$ with $x > 0$ is pairwise independent.

(Added the constraint $x>0$ since the pair $\{v,-v\}$ is obviously dependent.)

Answer 3

For e.g. $d=2$ the collection $\{(1,r)\mid r\in\mathbb R\}$ is pairwise independent.

Observe that equality $\lambda(1,r)+\mu(1,r')=(0,0)$ leads to $\lambda=\mu=0$ if $r\neq r'$.