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# 2.9 - 2.9 Dimension and Rank Denition Let B = cfw_b1 bp be...

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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

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bracketleftBigg 1 2 7 1 1 1 bracketrightBigg bracketleftBigg 1 2 7 0 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 = 0 0 1 , 1 1 0 H = Span 0 0 1 , 1 1 0 = x y z : x = y
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