Week4 - ECON 117 Mathematical Economics Spring 2008 Week 4...

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ECON 117 Spring 2008 Mathematical Economics Week 4 Def) Linear combination (broad meaning) Let { } { } 1 ,, im = xxx where . n i x \ If , then we call “ is a linear combination of 1 , m ii i i λλ = =⋅ yx \ y { } i x ” or “ y is linearly dependent on { } i x ”. Def) Linear dependence and independence The set of vectors { m } { } 1 ,, , n i = x \ , are linearly independent dependent ⎩⎭ if 1 1 only when 0 for all 1,2, , for some 0 m ii i i m i i = = ⎧⎫ ⋅= = = ⎪⎪ ⎨⎬ x0 . Ex) () ( ) 2 12 1 2 , 1,2 , 2,1 ∈= = xx x x \ Q) Show that and are linearly independent. 1 x 2 x Page 1 of 6
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ECON 117 Spring 2008 Mathematical Economics Q) Show that ( ) 3 3, 2 = x is linearly dependent on { } 12 , xx . Q) Show that { } 123 ,, xxx are linearly dependent. Theorem) Let { } { } 1 ,, , n in i =∈ x …\ . If { } i x are linearly independent, then can be represented by a linear combination of n ∀∈ y \ { } i x That is, n independent n-dimensional vectors span . n \ Q) Let ( ) ( ) 11 1 1 2 2 2 1 2 2 , , x == x . i.e., . Find the condition under which and are linearly dependent. 2 , \ 1 x 2 x Page 2 of 6
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ECON 117 Spring 2008 Mathematical Economics Ex) The condition for { } 123 ,, xxx , to be linearly dependent
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This note was uploaded on 09/15/2008 for the course ECON 324 taught by Professor Kim during the Spring '08 term at HKUST.

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Week4 - ECON 117 Mathematical Economics Spring 2008 Week 4...

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