examples152-ch7

examples152-ch7 - MATH 152 Fall 2006-07 Applied Linear...

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Unformatted text preview: MATH 152 Fall 2006-07 Applied Linear Algebra & Differential Equations Worked Examples Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology September 1, 2006 ii Contents Table of Contents iii 1 Introduction 1 2 First-Order Differential Equations 5 3 Second-Order Linear Equations 39 4 Laplace Transform 83 5 Matrix 127 6 Systems of Linear Equations 135 7 Euclidean Vector 155 8 Eigenvalue and Eigenvector 177 9 Systems of Differential Equations 209 10 Orthogonality 231 iii Chapter 7 Euclidean Vector ¥ Example 7.1 (Linear combination of vectors) Consider the vectors v 1 = (1 , 2 ,- 1) , v 2 = (2 , 3 ,- 1) , v 3 = (3 , 1 , 1) . Is b = (2 , 7 ,- 4) a linear combination of v 1 , v 2 , v 3 ? Solution By a linear combination, we mean c 1 v 1 + c 2 v 2 + c 3 v 3 = b for some c 1 ,c 2 ,c 3 ⇐⇒ c 1 2 4 1 2- 1 3 5 + c 2 2 4 2 3- 1 3 5 + c 3 2 4 3 1 1 3 5 = 2 4 2 7- 4 3 5 ⇐⇒ 8 < : c 1 +2 c 2 +3 c 3 = 2 , 2 c 1 +3 c 2 + c 3 = 7 ,- c 1- c 2 + c 3 =- 4 has solutions . Therefore we need to check whether the above system of linear equations has solutions for ( c 1 ,c 2 ,c 3 ) or not. By doing this, we simplify the augmented matrix as follows 2 4 1 2 3 2 2 3 1 7- 1- 1 1- 4 3 5- 2 R 1 + R 2------→ R 1 + R 3 2 4 1 2 3 2- 1- 5 3 1 4- 2 3 5 2 R 2 + R 1--------→ R 2 + R 3 ,- R 2 2 4 1- 7 8 1 5- 3- 1 1 3 5- 7 R 3 + R 1---------→ 5 R 3 + R 2 ,- R 3 2 6 4 / £ ¡ ¢ 1 1 / £ ¡ ¢ 1 2 / £ ¡ ¢ 1- 1 3 7 5 . Since the b-column is nonpivot, the system has solutions for ( c 1 ,c 2 ,c 3 ). In fact, the reduced row echelon form implies that we have b = (1) · v 1 + (2) · v 2 + (- 1) · v 3 = v 1 + 2 v 2- v 3 . Remark . In fact we may verify that the above expression of b in terms of v 1 , v 2 , v 3 is correct. v 1 + 2 v 2- v 3 = 2 4 1 2- 1 3 5 + 2 2 4 2 3- 1 3 5- 2 4 3 1 1 3 5 = 2 4 1 + 4- 3 2 + 6- 1- 1- 2- 1 3 5 = 2 4 2 7- 4 3 5 = b . 2 155 7. Euclidean Vector ¥ Example 7.2 (Linear combination of vectors) Consider the vectors v 1 = (1 , 2 ,- 1) , v 2 = (2 , 3 ,- 1) , v 3 = (3 , 1 , 2) . Is b = (2 , 7 ,- 4) a linear combination of v 1 , v 2 , v 3 ? Solution By a linear combination, we mean c 1 v 1 + c 2 v 2 + c 3 v 3 = b for some c 1 ,c 2 ,c 3 ⇐⇒ c 1 2 4 1 2- 1 3 5 + c 2 2 4 2 3- 1 3 5 + c 3 2 4 3 1 2 3 5 = 2 4 2 7- 4 3 5 ⇐⇒ 8 < : c 1 +2 c 2 +3 c 3 = 2 , 2 c 1 +3 c 2 + c 3 = 7 ,- c 1- c 2 +2 c 3 =- 4 has solutions . Therefore we need to check whether the above system of linear equations has solutions for ( c 1 ,c 2 ,c 3 ) or not. By doing this, we simplify the augmented matrix as follows 2 4 1 2 3 2 2 3 1 7- 1- 1 2- 4 3 5- 2 R 1 + R 2------→ R 1 + R 3 2 4 1 2 3 2- 1- 5 3 1 5- 2 3 5 R 2 + R 3-----→ 2 6 4 / £ ¡ ¢ 1 2 3 2 / £ ¡ ¢- 1- 5 3 / £ ¡ ¢ 1 3 7 5 ....
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examples152-ch7 - MATH 152 Fall 2006-07 Applied Linear...

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