Basis

Basis

Definition

Let be a subspace of vector space and let . We say that is a basis for if:

  1. is linearly independent

If , then we define to be the basis for

(see span)

Intuition

A basis for a subspace of is a linearly independent spanning set for . That is, it is the set of the minimum number of vectors that we can still use those vectors create every vector we want in a certain space, or span.

It then makes sense, that if we can express a vector as a linear combination of the other vectors, said vector is useless, and we don't need it.

Also see the Standard Basis below, which can clear things up too.

Theorem

Let . Then is a basis for if and only if has .

Intuition

A basis must be a square matrix, so we can the entire space the basis is meant to cover. It also must span the subspace it is trying to cover.

Example

Show that is a basis for

First, show that is linearly independent.

Consider the matrix , and carry it to REF

from which we see that , so is linearly independent. (See rank)

Since has rows and , by the system-rank theorem, the system is consistent for every , so by the "Span theorem", .
This shows that

Since , and is closed under linear combinations, we have that .

Hence, by Set Equality, , and so is a basis for .

Finding a Basis

There are some examples of finding the basis for linear transformations.

Generally, you want to solve the system, and then any vectors with free variables are your basis vectors.

Standard Basis

Definition

In , the standard basis is

which is analogous to the unit vectors and .

In :

In :

and so on.

Change of Basis

Question

Recall that apart from the standard basis, we can use any vectors as our basis, as long as they are not linear combinations of each other.

How do we translate between coordinate systems?

Example

Lets say we have an alternate basis and and the vector of that basis.

Well, all we're saying is that , so in terms of the standard basis,

But this is just the matrix-vector product:

Which is nothing but a matrix transformation whose result is the same as if you just used the alternate basis.

What about the other way?

Remember, if , we can invert the matrix to produce the opposite effect. That is, we can now translate from our standard basis to theirs. (since , then )

Change of Basis of a Linear Transformation

Steps

Given a matrix transformation in your basis:

  1. Transform the vector from the other basis to your basis
  2. Apply the transformation
  3. Transform the vector from your basis back to the other basis
Definition

Given a vector in an alternate basis, and a matrix transformation . Then the transformation of is