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

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

3. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB A basis for a vector space V is an independent set of vectors that span V. For example, in ℝ 2 the standard basis is Suppose we have a different set of independent vectors from ℝ 2. For example, consider the set Is this an alternate basis for ℝ 2 ?

4. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB A basis for a vector space V is an independent set of vectors that span V. For example, in ℝ 2 the standard basis is Suppose we have a different set of independent vectors from ℝ 2. For example, consider the set Is this an alternate basis for ℝ 2 ? The answer is YES. In fact any set of n independent vectors form a basis for ℝ n. 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:

5. Example: Consider the vector In the standard basis for ℝ 2, our vector can be written as: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. Example: Consider the vector In the standard basis for ℝ 2, our vector can be written as: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We would like to be able to write this vector in terms of our alternate basis

7. Example: Consider the vector In the standard basis for ℝ 2, our vector can be written as: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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.

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

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

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

11. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 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 P B.