wksht11sol - MATH 152 L1& L2 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 11 Worksheet Solutions : Eigenvalue and Eigenvector Q1. (Maximal independent vectors) Show that the five vectors in R 3 : v 1 = (1 , 2 , 1), v 2 = (- 1 , 1 , 0), v 3 = (3 ,- 2 , 4), v 4 = (0 ,- 2 ,- 3), v 5 = (2 ,- 3 , 4) are linearly dependent. Then find maximal number of linearly independent vectors from the five. Solution We simplify the matrix £ v 1 v 2 v 3 v 4 v 5 / as follows.   1- 1 3 2 2 1- 2- 2- 3 1 4- 3 4  - 2 R 1 + R 2------→- R 1 + R 3   1- 1 3 2 3- 8- 2- 7 1 1- 3 2   R 3 + R 1------→- 3 R 3 + R 2   1 4- 3 4- 11 7- 13 1 1- 3 2  - 1 11 R 2-----→ R 2 ↔ R 3   1 4- 3 4 1 1- 3 2 1- 7 / 11 13 / 11  - 4 R 3 + R 1------→- R 3 + R 2    / £ ¡ ¢ 1- 5 / 11- 8 / 11 / £ ¡ ¢ 1- 26 / 11 9 / 11 / £ ¡ ¢ 1- 7 / 11 13 / 11    . The five vectors are linearly dependent. In fact, v 1 , v 2 , v 3 form a set of maximal independent vectors and v 4 =- 5 11 v 1- 26 11 v 2- 7 11 v 3 , v 5 =- 8 11 v 1 + 9 11 v 2 + 13 11 v 3 . Q2. (Basis of R 3 ) Consider the matrix A =   1 1 2 1 1 3 1 1   . Find linearly independent vectors that span (a) Nul A , (b) Col A , (c) Row A . Solution We simplify the matrix A as follows....
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht11sol - MATH 152 L1& L2 Applied Linear Algebra and...

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