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Unformatted text preview: 2.9 Dimension and Rank Definition: Let B = { vector b 1 , , vector b p } be a basis for a sub space H R n . For all vectorx H , vectorx = c 1 vector b 1 + c 2 vector b 2 + + c p vector b p for some c 1 , ,c p R . We denote [ vectorx ] B = c 1 . . . c p the coordinates of vectorx with respect to the basis B . Note: Since vector b 1 , , vector b p are linear independent, these coor dinates are unique. Why? Thare are many ways to see this. For example: If A = [ vector b 1  ,  vector b p ], then the RE form U = CA has p pivots, so Avector c = vectorx has a unique soluntion. EX: B = braceleftBiggbracketleftBigg 1 1 bracketrightBigg , bracketleftBigg 2 1 bracketrightBiggbracerightBigg is a basis for R 2 b 1 b 2 vectorx = bracketleftBigg 7 1 bracketrightBigg Find [ vectorx ] B 1 bracketleftBigg 1 2 7 1 1 1 bracketrightBigg bracketleftBigg 1 2 7 3 6 bracketrightBigg bracketleftBigg 1 2 7 0 1 2 bracketrightBigg bracketleftBigg 1 0 3 0 1 2 bracketrightBigg [ vectorx ] B = bracketleftBigg 3 2 bracketrightBigg . EX: B = 1 , 1 1 H = Span 1...
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This note was uploaded on 04/09/2011 for the course MATH 232 taught by Professor Russel during the Spring '10 term at Simon Fraser.
 Spring '10
 Russel
 Linear Algebra, Algebra

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