Linear independence

So, it ends up that these vectors got together and wrote a declaration of independence and that's what we'll cover here. Just joking! We do need humor every so often in a math book. To explain linear independence, we need to go back to the concept of a linear combination that we introduced earlier in this book.

Linear combination

We learned in Chapter 2, Superposition with Euclid, that linear combinations are the scaling and addition of vectors. I would like to give a more precise definition as we go beyond three-dimensional space.

A linear combination for vectors |x1⟩,|x2⟩, … |xn⟩ and scalars c1, c2, … cn in a vector space, V, is a vector of the form:

(1)

Basically, it is still scaling and addition, but now we can do it for vectors of any dimension and with as many finite numbers of vectors as we wish.

Let's look at an example:

So now that we have defined linear combinations, let...