Let $\nu_1,...,\nu_n \in \mathbb{R}^n$ be vectors. Let $\mathcal{I}$ be the collection of subsets $I \subseteq \{1,2,...,k\}$ for which the $\nu_i, i\in I$ are affinely independent.
Exercise: I need to prove or disprove that $\mathcal{I}$ is the collection of independent sets of a matroid.
I know that the pair $M = (E,\mathcal{I})$ is a matroid if the properties
i) $\emptyset \in \mathcal{I}$.
ii) If $I\subseteq J$ and $J\in\mathcal{I}$, then $I\in\mathcal{I}$.
iii) If $I,J\in \mathcal{I}$ and $\left|I\right| <\left|J\right|$, then there is an element $e\in J\backslash I$ such that $I+e\in\mathcal{I}$.
all hold. In this case $I \in \mathcal{I}$ are called the independent sets.
So in order to solve the exercise I need to show that properties i),ii) and iii) all hold if $\mathcal{I}$ is the collection of subsets $I \subseteq \{1,2,...,k\}$ for which the $\nu_i, i\in I$ are affinely independent.
What I've tried: I had to look up affine independence for this exercise and the most comprehensible definition I came across was this one:
The points $x_k$ are affine independent when $$ \sum \lambda_k x_k = 0 \text{ with }\sum \lambda_k =0 $$ implies all $\lambda_k = 0$.
I have no idea how to use this definition though. Let's say I want to show that $\emptyset\in \mathcal{I}$ (or $\emptyset \notin\mathcal{I}$). How do I show that $\emptyset$ is affinely independent or affinely dependent? And if I need to show that several vectors are affinely independent, do I look at all the $\sum\lambda_k\nu_k$? Since points in $\mathbb{R}^n$ are given by vectors right?
Questions:
How do I use affine independence to show that $\emptyset\in\mathcal{I}$ or $\emptyset\notin\mathcal{I}$?
Take vectors $\nu_i$ for some $I$. How do I show that $\nu_i$ are affinely (in)dependent?
(optional) How do I solve this exercise?
Thanks in advance!
When you're working with the empty set, the two sums in your definition of affine independence are the empty sum. By convention, this is zero.
You don't strictly need this to do the exercise, but it's a good question if you want to play with some examples. If $M$ is the matrix containing your affine vectors as columns, then $M$ is affinely independent if and only if $M$ with a row of all $1$s is linearly independent. (It's a small exercise to show that this is equivalent to the definition that you gave.) So you can use the same techniques that you use to show linear (in)dependence.