Algebra Lineal Friedberg 4 Ed Solucion Caps 1-4

LINEAR ALGEBRA
4th STEPHEN H.FRIEDBERG, ARNOLD J.INSEL, LAWRENCE E.SPENCE

Exercises Of Chapter 1-4

http : //math.pusan.ac.kr/caf e home/chuh/

Contents
§1. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. Introduction . . . . . . . . . . . . . . . . . . . . . . . . Vector Spaces . . . . . .. . . . . . . . . . . . . . . . . Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . Linear Combinations and Systems of Linear Equations Linear Dependence and Linear Independence . . . . . . Bases and Dimension . . . . . . . . . . . . . . . . . . . Maximal Linearly Independent Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 13 9 19 24 32 50

§2. Linear Transformations and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. Linear Transformations, Null Spaces, and Ranges . . . . . . . . . The matrix representation of a linear transformation . . . . . . . Composition of Linear Transformations and Matrix Multiplication Invertibility and Isomorphisms . . . . . . . . . . . . .. . . . . . . The change of Coordinate Matrix . . . . . . . . . . . . . . . . . . Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous Linear Differential Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 77 87 101 116 123 140

§3. Elementary Matrix Operations and Systems of Linear Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.1. 3.2. 3.3. 3.4. Elementary Matrix Operations and Elementary Matrices The Rank of a Matrix and Matrix Inverse . . . . . . . . Systems of Linear equations - Theoretical aspects . . . . Systems of Linear equations - Computational aspects . . . . . . . . . . . . . . . . . . . . . . 154 162 172 178§4. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.1. 4.2. 4.3. 4.4. 4.5. Determinants of Order 2 . . . . . . . . . . . . . Determinants of Order n . . . . . . . . . . . . . Properties of Determinants . . . . . . . . . . . . Summary-Important Facts about Determinants A characterization of the Determinants . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 192 198 213 215

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 = 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 = 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)

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1.1. 4. x = (a1 , a2 , · · · , an ) ∈ Rn , i = 1, 2,· · · , n 0 = (0, 0, · · · , 0) ∈ Rn s.t. x + 0 = x, ∀x ∈ Rn

5. x = (a1 , a2 ) ⇒ tx = t(a1 , a2 ) = (ta1 , ta2 )

6. A + B = (a + c, b + d), M = ( a+c , b+d ) 2 2

7. C = (v − u) + (w − u) + u = v + w − u − → − → −→ 1 −→ − − OD + 2 DB = OA + 1 AC 2
1 i.e. w + 1 (v − w) = 1 (v + w) = u + 2 (v + w − u − u) 2 2

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1.2.

1.2. Vector Spaces
1. (a) T (b) F (If ∃0 s.t. x + 0 =x, ∀x ∈ V, then 0 = 0 + 0 = 0 + 0 = 0, ∴ 0 = 0) (c) F (If x = 0, a = b, then a · 0 = b · 0 but a = b) (d) F (If a = 0, x = y, then a · x = 0 = a · y but x = 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 degf =degg=1, deg(f + g)=0) (i) T (p.10 Example4) (j) T (k) T (p.9 Example3)  0 0 2.  0 0 0 0 0 0  0 0  0 0

3. M13 =...