BASED

GUESS WHAT WE’RE DOING TODAY!

…

THATS RIGHT! MORE LINEAR ALGEBRA! WOOHOO!

So,

Here we go,

This is it. This is the big one. For all the marbles. What we’ve been leading up to for the past 4 blogs.

Vector Bases.

DUN DUN DUN!

I’m mostly joking in a distinctly unfunny way here. These actually aren’t that important, but they can be reasonably powerful tools and they represent a good synthesis of the stuff we’ve covered so far.

We’ll start with a definition and then a theorem [1]:

Def – A basis is a set of linearly independent vectors that span a vector space.

Thrm – Every vector space has a basis.

Basically, a basis can be used to describe a given vector space. We take a set of independent vectors that can be used in a linear combination to make any other vector in that space. For example:

What is the basis of the vector space in R3 [2], V = {: 2x + y = 0}?

What we’re looking for here is a set of vectors that can be multiplied and added to make any vector that fits that description, so a good starting place is to figure out what vectors actually fit that description.

We can see that vectors in this space are 3D and must have values x and y such that 2x + y = 0. A little bit of algebra can show us that y = -2x and we can say that the vectors in this space are of the form . This vector is arbitrary because it has 2 variables that could be any number so it doesn’t have defined values. To find the linear combination that constructs this vector we can separate the parts that don’t share the same variable: + <0, 0, z>, and then remove the variables x * <1, -2, 0> + z * <0, 0, 1> and WHADDAYA KNOW BA-DA-BING BA-DA-BOOM THAT LOOKS TO ME LIKE A LINEAR COMBINATION OF VECTORS THAT CAN REPRESENT ANY VECTOR IN OUR DEFINED VECTOR SPACE! WE’VE FOUND OUR BASIS!

Specifically, the basis is the set B = {<1, -2, 0>, <0, 0, 1>}.

If you’d like to check, you can verify for yourself that it is infact possible to make any vector in V by multiplying and adding the ones in B. I’m not sure if you really need to because most of what we did here is algebra, but maybe I didn’t explain it well enough to convince you. Maybe you don’t even know how to check because I didn’t explain that well enough.

Anyways,

I’ll probably talk about why these bases are cool or something next week.

[1] – Theorems are just like important rules that you should know. They are usually proven when you’re given them, but we’re too cool to show our work here.

[2] – R3 is just the 3D space of real numbers, think of it like a 3D coordinate plane except instead of putting functions on it, we’re putting vectors.