# bases - Change of Basis Matrices A basis for a vector space...

This preview shows pages 1–2. Sign up to view the full content.

Change of Basis Matrices A basis for a vector space is not unique. In fact, given a vector space V and two different bases B = < -→ v 1 , . . . , -→ v n > and C = < -→ w 1 , . . . , -→ w n > we can easily construct a linear map that sends each basis vector in B to a different basis vector in C by letting -→ v j 7→ -→ w j and extending linearly. The purpose of this handout is to illustrate the routine for finding the matrix representation for that linear map in the case where our vector space V = R n . 1 The Easy Direction – from B to E n Recall the notation for the standard basis in R n is E n = * 1 0 . . . , 0 1 . . . , . . . + . Now suppose that B = < -→ v 1 , . . . , -→ v n > is any other basis for R n . We may represent any vector in R n in terms of B using the notation a 1 a 2 . . . a n B := a 1 -→ v 1 + · · · + a n -→ v n . We’ll assume that the vectors a 1 a 2 . . . a n B and c 1 c 2 . . . c n represent the same vector in R n but written relative to the two given bases. (Think of this as these vectors have the same picture – magnitude and direction.) To change a vector in terms of the basis B to a vector in terms of E n we need only do the following matrix multiplication ( -→ v 1 · · · -→ v n ) a 1 a 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern