Friedberg Solutions

Friedberg Solutions - LINEAR ALGEBRA 4th STEPHEN...

This preview shows pages 1–8. Sign up to view the full content.

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/

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

View Full Document
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
§ 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

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

View Full Document
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
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 XT] v k h i j t V W X XT^ 2 PNU-MATH

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

View Full Document
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
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 .

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.

This note was uploaded on 07/16/2011 for the course ECON 001 taught by Professor Ana during the Spring '11 term at Allen SC.

Page1 / 225

Friedberg Solutions - LINEAR ALGEBRA 4th STEPHEN...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online