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Macroeconomics Exam Review 105

Macroeconomics Exam Review 105 - c 2001 Michael Carter All...

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This implies that x can also be expressed as a linear combination of elements in { x 1 , 𝑥 2 , ..., x 𝑘 } , that is there exist scalars 𝛼 1 , 𝛼 2 , . . . , 𝛼 𝑘 such that x = 𝑘 𝑖 =1 𝛼 𝑖 x 𝑖 or x = 𝑘 𝑖 =1 𝛼 𝑖 x 𝑖 = 𝑛 𝑖 = 𝑘 +1 𝛼 𝑖 x 𝑖 which contradicts the assumption that 𝐵 is a basis for 𝑋 . Therefore { 𝑓 ( x 𝑘 +1 ) , . . . , 𝑓 ( x 𝑛 ) } is a basis for 𝑓 ( 𝑋 ) and therefore dim 𝑓 ( 𝑥 ) = 𝑛 𝑘 . We conclude that dim kernel 𝑓 + dim 𝑓 ( 𝑋 ) = 𝑛 = dim 𝑋 3.25 Equation (3.2) implies that nullity 𝑓 = 0, and therefore 𝑓 is one-to-one (Exercise 3.18). 3.26 Choose some x = ( 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑛 ) 𝑋 . x has a unique representation in terms of the standard basis (Example 1.79) x = 𝑛 𝑗 =1 𝑥 𝑗 e 𝑗 Let y = 𝑓 ( x ). Since 𝑓 is linear y = 𝑓 ( x ) = 𝑓 𝑛 𝑗 =1 𝑥 𝑗 e 𝑗 = 𝑛 𝑗 =1 x 𝑗 𝑓
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