In one of my lectures, after defining linear dependence as such:
Let R be a ring and M an R-module. A finite family $(x_i)_{i=1, …, n}$ of elements in M are called linearly dependent if there exists $(a_i)_{i=1, …, n}$ in R, not all zero, such that $a_1x_1 + … +a_nx_n = 0$.
They followed it with a remark:
If a family contains repetition, it is linearly dependent: Indeed, if $x_{i_0}=x_{i_1}$ with $i_0<i_1$ then take $a_i$ such that
- $a_i=0 \text{ for } I \notin \{i_0, i_1\}$
- $a_{i_0} = 1 \text{ and } a_{i_1} = -1$
Together with the above definition, this makes sense mathematically. However, intuitively it does not, since I imagine two linearly dependent vectors in, say for example, 2 dimensions as on the same line. With this, if you have two linearly independent vectors, and just add the first one again to the family, suddenly they're all linearly dependent? How does this make sense intuitively?
I think my confusion here lies with the concept of a "family". Since in a set there are no repeats, this problem wouldn't occur. So why is linear dependence different for a set and a family, if they have the same elements and the family only has a few repeats?
Best Answer
I think you're thinking of linear dependence / independence as a property of the individual vectors, but it's a property of the family of vectors.
If you've got a bowl of M&Ms that are all different colors-- say, red, blue, and yellow-- you might call it an "all different colors" bowl of M&Ms. But if you put another blue M&M in the bowl, now you don't have an "all different colors" bowl of M&Ms anymore. It's not that the individual M&Ms now lack the "all different colors" property, it's that the bowl does.
Similarly, if you've got a family of linearly independent vectors, and you add a duplicate, it's not the vectors that become linearly dependent, it's the family of vectors that becomes linearly dependent.