1. Coordinate Systems Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. For example, in ℝ2 the standard basis is
A basis for a vector space V is an independent set of vectors that span V.
For example, in ℝ2 the standard basis is
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. For example, in ℝ2 the standard basis is
A basis for a vector space V is an independent set of vectors that span V.
For example, in ℝ2 the standard basis is
Suppose we have a different set of independent vectors from ℝ2.
For example, consider the set
Is this an alternate basis for ℝ2?
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. For example, in ℝ2 the standard basis is
A basis for a vector space V is an independent set of vectors that span V.
For example, in ℝ2 the standard basis is
Suppose we have a different set of independent vectors from ℝ2.
For example, consider the set
Is this an alternate basis for ℝ2?
The answer is YES. In fact any set of n independent vectors form a basis for ℝn.
Our task is to be able to switch back and forth between different bases for the same vector space.
An example should help clarify the situation:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. In the standard basis for ℝ2, our vector can be written as:
Example:
Consider the vector
In the standard basis for ℝ2, our vector can be written as:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. In the standard basis for ℝ2, our vector can be written as:
Example:
Consider the vector
In the standard basis for ℝ2, our vector can be written as:
We would like to be able to write this vector in terms of our alternate basis
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. In the standard basis for ℝ2, our vector can be written as:
Example:
Consider the vector
In the standard basis for ℝ2, our vector can be written as:
We would like to be able to write this vector in terms of our alternate basis
An equation would look something like this:
Of course this type of equation should look familiar. It is just a 2x2 linear system. We have solved this type of equation many times.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. We can solve this by finding the inverse of the coefficient matrix.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. We can solve this by finding the inverse of the coefficient matrix.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. We can solve this by finding the inverse of the coefficient matrix.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. You may have noticed that the matrix we used was comprised of the basis vectors in our alternate basis. Here is the notation they use in the textbook.
In order to convert from the standard basis to the alternate basis, we found the inverse of PB.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB