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Linear Equations • • • • • • 1.1 1.2 1.3 1.4 1.5 1.7 System of linear Equations Row Reduction and Echelon Forms Vector Equations 向量方程 The Matrix Equation Ax = b 矩阵方程 Solution Sets of Linear Systems Linear Independence 线性无关 • Solution 1 2 7 x1 2 x2 5 4 5 6 3 1 2 7 1 0 3 2 5 4 ~ 0 1 2 5 6 3 0 0 0 x1 2 x2 7 2 x1 5 x2 4 5 x1 6 x2 3 b = 3a1+ 2a2 x1 2 x2 x3 4 5 x2 3x3 1 1 2 1 4 0 x1 5 x2 3 x3 1 Ax b x1 1 2 1 4 0 5 3 x2 1 x 3 a matrix equation • a. b. c. d. THEOREM 4 The following statements are logically equivalent. For each b in Rm, the equation Ax = b has a solution. Each b in Rm is a linear combination of the column of A. The columns of A span Rm. A has a pivot position in every row. 10 x1 3x2 2 x3 0 x1 0.3x2 0.2 x3 0.3 0.2 x 1 x 0 x2 10 3 2 x2 0 x 2 3 x3 0 1 x3 The solution set of Ax=0 can always be expressed explicitly as Span{v1, v2, … , vp} for suitable vectors v1, v2, … , vp. 1 0 4 / 3 1 [ A, b] 0 1 0 2 0 0 0 0 x1 1 43 x3 1 43 x3 1 43 Ax b x x2 2 2 0 2 x3 0 1 x3 x3 0 x3 0 • THEOREM 6 Suppose Ax = b is consistent, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + vh , where vh is any solution of the homogeneous equation Ax = 0. 1.7 Linear Independence 线性无关 • DEFINITION • A set {v1, v2, … , vp} in Rn is said to be linearly independent if the vector equation x1v1+ x2v2+ … + xpvp=0 has only the trivial solution. • The set {v1, v2, … , vp} is said to be linearly dependent if there exist weights c1, c2, …, cp, not all zero, such that c1v1+ c2v2+ … + cpvp=0. • A set is linearly dependent if and only if it is not linearly independent. • 不是“线性独立” • 没有“线性有关”这个词,应该是“线性 相关” • Linear dependence relation, 线性相关关系 c1v1+ c2v2+ … + cpvp=0 • linearly dependent set, 线性相关组 {v1, v2, … , vp} is linearly dependent • linearly independent set 线性无关组 {v1, v2, … , vp} is linearly independent • EXAMPLE 1 1 • Let 4 2 v1 2 , v2 5 , v3 1 . 5 6 0 a. Determine if the set {v1, v2, v3} is linearly independent. b. If possible, find a linear dependence relation among v1, v2, v3. DOES x1v1+ x2v2+ x3v3=0 has only the trivial solution? 2 1 4 2 1 4 [v1 , v2 , v3 ] 2 5 1 ~ 0 3 3 5 6 0 0 14 10 1 4 2 1 4 2 1 4 2 [v1 , v2 , v3 ] 2 5 1 ~ 0 3 3 ~ 0 1 1 3 6 0 0 6 6 0 0 0 1 0 2 2v1-v2+v3=0 ~ 0 1 1 0 0 0 Linear Independence of Matrix Columns The columns of a matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution. Ax=0 A1x1+ A2x2+…+ Anxn+=0 • EXAMPLE 2 Determine if the column of the matrix 0 1 4 A 1 2 -1 5 8 0 are linearly independent. 1 A ~ 0 5 1 ~ 0 0 2 1 1 4 8 0 2 1 1 4 -2 5 Sets of One or Two Vectors • A set containing only one vector – say, v – is linearly independent if and only if v is not the zero vector. • A set of two vectors {v1, v2} is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other. Sets of Two or More Vectors • THEOREM 7 • Characterization of Linearly Dependent Sets • A set S={v1, v2,…, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. Sets of Two or More Vectors • A set S={v1, v2,…, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. :The set {v1, v2, … , vp} is linearly dependent , so there exist weights c1, c2, …, cp, not all zero, such that c1v1+ c2v2+ … + cpvp=0. c j 0, c j v j c1v1 c2 v2 vj c1 c v1 2 v2 cj cj c j 1 cj c j 1v j 1 c j 1v j 1 v j 1 c j 1 cj v j 1 c pv p cp cj vp Sets of Two or More Vectors • THEOREM 7 • Characterization of Linearly Dependent Sets • A set S={v1, v2,…, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. If S is linearly dependent and v10,then some vj is a linear combination of the preceding vectors, v1, v2,…, vj-1. • THEOREM 8 • If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. • That is, any set {v1, v2,…, vp} in Rn is linearly dependent if p>n. x1v1+ x2v2+ … + xpvp=0 [v1 , v2 , , v p ] • THEOREM 9 • If a set S={v1, v2,…, vp} in Rn contains the zero vector, then the set is linearly dependent. 50+ x1v1+ x2v2+ … + xpvp=0 • Homework: Exercise1.7 2, 6, 12, 32