Math 313 - Understanding a Basis

Dear Math 313 and 302 students,


I know how difficult it can be to understand the concept of a basis. Hopefully this will help.

A basis is simply the set containing the fewest necessary vectors possible to represent a space.

If that didn't make any sense at all, then picture a blank x-y axis. Let the two lines (the x-axis and the y-axis) be vectors. You can notice two things:

1. These two vectors are all you ever need to represent all of 2-dimensional space
   (Think about it-- all you need an x-value and a y-value and you can represent any point)
   
2. The two vectors are linearly independent.
   (Try and represent the point (0,5) with only the x-axis - kinda hard to do, right?)

Therefore these two vectors represent a basis for 2-dimensional space.


Now, let's generalize this to 3-dimensional space. Imagine the axes for 3-d space. Now let those axes be vectors. Again:

1. The three vectors are all you ever need to represent all of 3-d space (x,y,z)
2. The vectors are linearly independent.

Therefore these three vectors represent a basis for 3-dimensional space.


Are you seeing the pattern? Now let's pretend that in 3-d space, you didn't have the z-axis but had a vector from (0,0,0) to (1,1,0). Do the x-axis, y-axis, and the new vector form a basis for 3-dimensional space?

If you answered no, then you'd be correct. Why?

Can you represent the point (0,0,1)? Nope-- the vectors are linearly dependent, so you can't represent all of 3-dimensional space. Therefore the vectors do not form a basis for 3-d space.

Got it yet? I hope so. As a final reminder, the requirements to be a basis for, say, n-dimensional space are:

1. You must have n n-dimensional vectors (n vectors with n entries)
2. Those vectors must be linearly independent

If your vectors fulfill that requirement, you've got yourself a basis!


-Math Lab Blogger