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

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Unformatted text preview: 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:....
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week6 - Row Equivalence of matrices March 1 2007 0.1 Row...

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