week6 - Row Equivalence of matrices March 1 2007 0.1 Row...

• Notes
• 2

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

Row Equivalence of matrices March 1, 2007 0.1 Row equivalence Let F be a field and let m and n be positive integers. Two m by n matrices are said to be row equivalent if there is an invertible matrix S such that B = SA . (Check that this is indeed an equivalence relation.) The textbook shows that any two row equivalent matrices have the same null space. In fact the converse is also true, so that we have the following theorem: Theorem: If A and B are two m by n matrices, then the following conditions are equivalent: 1. There exists an invertible matrix S such that B = SA 2. The matrices A and B have the same nullspace. This theorem can be translated into a statement about linear transfor- mations by considering the linear transformations L A , L B , and L S associated with the matrices above. It seems more revealing to treat this more abstract looking version. Theorem Let V and W be finite dimensional vector spaces, and let T and T be linear transformations from V to W . Then the following are equivalent: 1. There exists an isomorphism

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

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