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

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
ECON 117 Spring 2008 Mathematical Economics Ex) The condition for { } 123 ,, xxx , to be linearly dependent

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online