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:
- 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) - The two vectors are linearly independent.
(Try and represent the point (0,5) with only the x-axis - kinda hard to do, right?)
Now, let's generalize this to 3-dimensional space. Imagine the axes for 3-d space. Now let those axes be vectors. Again:
- The three vectors are all you ever need to represent all of 3-d space (x,y,z)
- The vectors are linearly independent.
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:
- You must have n n-dimensional vectors (n vectors with n entries)
- Those vectors must be linearly independent
-Math Lab Blogger
