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# 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 Definition. An ordered basis ( b 1 , b 2 , . . . , b n ) is an “ordered set” of vec- tors which is a basis for some vector space. Definition 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 v relative to B , denoted v B . Example. Page 211 number 6. Note. To find v B : (1) write the basis vectors as column vectors to form [ b 1 , b 2 , . . . , b n | v ], (2) use Gauss-Jordan elimination to get [ I | v B ] .

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3.3 Coordinatization of Vectors 2 Definition. 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 ( v 1 + v 2 ) = α ( v 1 ) + α ( v 2 ) , and (2) if v V and r R then α ( rv ) = ( 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.
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c3s3 - 3.3 Coordinatization of Vectors 1 Chapter 3 Vector...

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