lect136_12_w09

lect136_12_w09 - Friday, January 30 - Lecture 12 : Rowspace...

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Friday, January 30 Lecture 12 : Rowspace of a matrix A . (Refers to section 3.4) Expectations : 1. Define row space of a matrix. A . 2. Find a spanning family for the rowspace of A where the size of the spanning family equals rank A . 12.1 . Definitions Let A be an m × n matrix. Let { r 1 , r 2 , ..., r m } be the m row vectors of the the m × n matrix A . We define, the rowspace of A , denoted by Row( A ) : = span( r 1 , r 2 , ..., r m ) in R n . 12.1 . 1 Observation Row( A ) is the span of a set of vectors; it is not just a subset of R n , it is a subspace of R n . 12.2 . Theorem Suppose A and B are two matrices where B is obtained from A by applying a single elementary row operation. The Row( A ) = Row( B ). Proof: If B is obtained from A by applying P ij or cR i , we clearly see that every row of B is a linear combination of rows of A. So the rows of B form a subset of Row( A ).
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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lect136_12_w09 - Friday, January 30 - Lecture 12 : Rowspace...

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