Why if $k>d+1$ then $\{x_j-x_1\}_{j=2}^k\subset \mathbb R^d$ is linearly dependent?

Question

I don't get this step in proof of Carathéodory's theorem (convex hull) Why:

Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points $x_2 − x_1, ..., x_k − x_1$ are linearly dependent

Why is this true?

How can we cay these points are linearly dependent?

Answer 1

Note that $k > d+1$, and that our points are vectors in $\Bbb R^d$. In $\Bbb R^d$ (or any $d$-dimensional vector space), any set consisting of more than $d$ vectors is linearly dependent.

Answer 2

What is the cardinality of $\{x_2 − x_1, ..., x_k − x_1\}$? Now remember that there aren't any linearly independent set of cardinality greater than $d$ in $\Bbb R^d$.