m235notes17 - Notes 17: Bases, Coordinates, Matrices...

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Unformatted text preview: Notes 17: Bases, Coordinates, Matrices Lecture November, 2011 Let V,W be vector spaces whose dimension is finite. Let F : V- W be a linear map. We show how to assosciate to F a matrix. We can not do this if we are only given the map F . We need more information. To do this we need to have bases for V and W . We begin by reviewing how we find matrices of linear maps when our vector spaces are R m and R n . Let e i denote the vector of length m all of whose entries are zero, except in the i-th postion. The entry in the i position is 1 . Problem: Let M be a linear map R m- R n . How do we construct a matrix M that induces the same function as M ? Observation 1. Let v 1 ,v 2 , v m be m vectors in R n ; then the matrix whose columns are v 1 , ,v m maps e i to v i for i = 1 i = m. Observation 2. Let F : R m- R n be a linear map. Then F is determined by the vectors F ( e i ) R n . Proof. Let x R m . We can write x = i = m i =1 x i e i . Then F ( x ) = F ( i = m X i =1 x i e i ) = i = m X i =1 x i F ( e i ) ....
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m235notes17 - Notes 17: Bases, Coordinates, Matrices...

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