c3s3 - 3.3 Coordinatization of Vectors 1 Chapter 3. Vector...

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3.3 Coordinatization of Vectors 1 Chapter 3. Vector Spaces 3.3 Coordinatization of Vectors Defnition. An ordered basis ( ~ b 1 , ~ b 2 ,..., ~ b n ) is an “ordered set” of vec- tors which is a basis for some vector space. Defnition 3.8. If B =( ~ b 1 , ~ b 2 ~ b n ) is an ordered basis for V and ~v = r 1 ~ b 1 + r 2 ~ b 2 + ··· + r n ~ b n , then the vector [ r 1 ,r 2 ,...,r n ] R n is the coordinate vector of relative to B , denoted B . Example. Page 211 number 6. Note. To Fnd B : (1) write the basis vectors as column vectors to form [ ~ b 1 , ~ b 2 ~ b n | ], (2) use Gauss-Jordan elimination to get [ I| B ] .
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3.3 Coordinatization of Vectors 2 Defnition. An isomorphism between two vector spaces V and W is a one-to-one and onto function α from V to W such that: (1) if ~v 1 ,~v 2 V then α ( 1 + 2 )= α ( 1 )+ α ( 2 ) , and (2) if V and r R then α ( r~v ( ) . If there is such an α ,then V and W are isomorphic , denoted V = W . Note. An isomorphism is a one-to-one and onto linear transformation. Theorem. The Fundamental Theorem o± Finite Dimensional Vectors Spaces.
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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c3s3 - 3.3 Coordinatization of Vectors 1 Chapter 3. Vector...

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