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Friedberg Solutions

Friedberg Solutions - LINEAR ALGEBRA 4th STEPHEN...

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LINEAR ALGEBRA 4th STEPHEN H.FRIEDBERG, ARNOLD J.INSEL, LAWRENCE E.SPENCE Exercises Of Chapter 1-4 http : //math.pusan.ac.kr/cafe home/chuh/
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Contents § 1. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4. Linear Combinations and Systems of Linear Equations . . . . . . 19 1.5. Linear Dependence and Linear Independence . . . . . . . . . . . . 24 1.6. Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7. Maximal Linearly Independent Subsets . . . . . . . . . . . . . . . 50 § 2. Linear Transformations and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1. Linear Transformations, Null Spaces, and Ranges . . . . . . . . . 55 2.2. The matrix representation of a linear transformation . . . . . . . 77 2.3. Composition of Linear Transformations and Matrix Multiplication 87 2.4. Invertibility and Isomorphisms . . . . . . . . . . . . . . . . . . . . 101 2.5. The change of Coordinate Matrix . . . . . . . . . . . . . . . . . . 116 2.6. Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.7. Homogeneous Linear Differential Equations with Constant Coeffi- cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 § 3. Elementary Matrix Operations and Systems of Linear Equa- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.1. Elementary Matrix Operations and Elementary Matrices . . . . . 154 3.2. The Rank of a Matrix and Matrix Inverse . . . . . . . . . . . . . 162 3.3. Systems of Linear equations - Theoretical aspects . . . . . . . . . 172 3.4. Systems of Linear equations - Computational aspects . . . . . . . 178
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§ 4. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.1. Determinants of Order 2 . . . . . . . . . . . . . . . . . . . . . . . 188 4.2. Determinants of Order n . . . . . . . . . . . . . . . . . . . . . . . 192 4.3. Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . 198 4.4. Summary-Important Facts about Determinants . . . . . . . . . . 213 4.5. A characterization of the Determinants . . . . . . . . . . . . . . . 215
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1.1. § 1. Vector Spaces 1.1. Introduction 1 . Only the pairs in (b) and (c) are parallel (a) x = (3 , 1 , 2) and y = (6 , 4 , 2) @ 0 6 = t R s.t. y = tx (b) (9 , - 3 , - 21) = 3( - 3 , 1 , 7) (c) (5 , - 6 , 7) = - 1( - 5 , 6 , - 7) (d) x = (2 , 0 , - 5) and y = (5 , 0 , - 2) @ 0 6 = t R s.t. y = tx 2 . (a) x = (3 , - 2 , 4) + t ( - 8 , 9 , - 3) (b) x = (2 , 4 , 0) + t ( - 5 , - 10 , 0) (c) x = (3 , 7 , 2) + t (0 , 0 , - 10) (d) x = ( - 2 , - 1 , 5) + t (5 , 10 , 2) 3 . (a) x = (2 , - 5 , - 1) + s ( - 2 , 9 , 7) + t ( - 5 , 12 , 2) (b) x = ( - 8 , 2 , 0) + s (9 , 1 , 0) + t (14 , - 7 , 0) (c) x = (3 , - 6 , 7) + s ( - 5 , 6 , - 11) + t (2 , - 3 , - 9) (d) x = (1 , 1 , 1) + s (4 , 4 , 4) + t ( - 7 , 3 , 1) 1 PNU-MATH
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1.1. 4 . x = ( a 1 , a 2 , · · · , a n ) R n , i = 1 , 2 , · · · , n 0 = (0 , 0 , · · · , 0) R n s.t. x + 0 = x, x R n 5 . x = ( a 1 , a 2 ) tx = t ( a 1 , a 2 ) = ( ta 1 , ta 2 ) 6 . A + B = ( a + c, b + d ) , M = ( a + c 2 , b + d 2 ) 7 . C = ( v - u ) + ( w - u ) + u = v + w - u --→ OD + 1 2 --→ DB = -→ OA + 1 2 -→ AC i.e. w + 1 2 ( v - w ) = 1 2 ( v + w ) = u + 1 2 ( v + w - u - u ) v hO¡S¢P iO£S¤P t d d XT] v k h i j t XT^ 2 PNU-MATH
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1.2. 1.2. Vector Spaces 1 . (a) T (b) F (If 0 0 s.t. x + 0 0 = x, x V, then 0 0 = 0 + 0 0 = 0 0 + 0 = 0 , 0 0 = 0) (c) F (If x = 0 , a 6 = b , then a · 0 = b · 0 but a 6 = b ) (d) F (If a = 0 , x 6 = y , then a · x = 0 = a · y but x 6 = y ) (e) T (f) F (An m × n matrix has m rows and n cilumns) (g) F (h) F (If f ( x ) = ax + b, g ( x ) = - ax + b , then deg f =deg g =1, deg( f + g )=0) (i) T (p.10 Example4) (j) T (k) T (p.9 Example3) 2 . 0 0 0 0 0 0 0 0 0 0 0 0 3 . M 13 = 3 , M 21 = 4 , M 22 = 5 4 . (a) 6 3 2 - 4 3 9 3 PNU-MATH
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1.2. (b) 1 - 1 3 - 5 3 8 (c) 8 20 - 12 4 0 28 (d) 30 - 20 - 15 10 - 5 40 (e) 2 x 4 + x 3 + 2 x 2 - 2 x + 10 (f) - x 3 + 7 x 2 + 16 (g) 10 x 7 - 30 x 4 + 40 x 2 - 15 x (h) 3 x 5 - 6 x 3 + 12 x + 6 5 . U = 8 3 1 3 0 0 3 0 0 , D = 9 1 4 3 0 0 1 1 0 , T = 17 4 5 6 0 0 4 1 0 ( * ) The total number of crossings Fall Spring Winter Brook trout 17 4 5 Rainbow trout 6 0 0 Brown trout 4 1 0 6 . M = 4 2 1 3 5 1 1 4 3 1 2 6 , 2 M - A = 3 1 1 4 4 0 1 3 5 2 1 9 35 suites were sold during the June sale 4 PNU-MATH
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1.2.
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