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SYSTEMS OF LINEAR EQUATIONS A linear equation A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables — say, x1,…., xn. A solution of the system is a list (s1, s2,…, sn) of numbers that makes each equation a true statement when the values s1,…, sn are substituted for x1,…, xn, respectively. The set of all possible solutions is called the solution set of the linear system. Slide 1.1- 1 SYSTEMS OF LINEAR EQUATIONS A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many solutions. Matrix notation 1. coefficient matrix 2. augmented matrix Solving systems of linear equations Slide 1.1- 2 SYSTEMS OF LINEAR EQUATIONS Elementary row operations include the following: 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant. Slide 1.1- 3 Row Reduction and Echelon Forms A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros. Slide 1.2- 4 Row Reduction and Echelon Forms If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form): 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. • A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position. Slide 1.2- 5 Row Reduction and Echelon Forms The row reduction algorithm Solutions of linear systems Existence and uniqueness theorem Slide 1.2- 6 VECTOR EQUATIONS A matrix with only one column is called a column vector, or simply a vector. 2 © 2012 Pearson Education, Inc. Slide 1.3- 7 VECTOR EQUATIONS The vector whose entries are all zero is called the zero vector and is denoted by 0. n For all u, v, w in and all scalars c and d: (i) u v v u (ii) (u v) w u (v w) (iii) u 0 0 u u (iv) u ( u) u u 0, where u denotes (1)u (v) c(u v) cu cv (vi) (c d )u cu du © 2012 Pearson Education, Inc. Slide 1.3- 8 VECTOR EQUATIONS (vii) c(du)=(cd)(u) (viii) 1u u n Given vectors v1, v2, ..., vp in and given scalars c1, c2, ..., cp, the vector y defined by y c1v1 ... c p v p is called a linear combination of v1, …, vp with weights c1, …, cp. © 2012 Pearson Education, Inc. Slide 1.3- 9 VECTOR EQUATIONS A vector equation x1a1 x2a 2 ... xna n b has the same solution set as the linear system whose augmented matrix is ----(5) a b. a a 1 2 n In particular, b can be generated by a linear combination of a1, …, an if and only if there exists a solution to the linear system corresponding to the matrix (5). © 2012 Pearson Education, Inc. Slide 1.3- 10 LINEAR COMBINATIONS Span: If v1, …, vp are in , then the set of all linear combinations of v1, …, vp is denoted by Span {v1, …, n vp} and is called the subset of spanned (or generated) by v1, …, vp. That is, Span {v1, ..., vp} is the collection of all vectors that can be written in the form n c1v1 c2 v2 ... c p v p with c1, …, cp scalars. Geometric description © 2012 Pearson Education, Inc. Slide 1.3- 11 MATRIX EQUATION Ax b Definition: If A is an m n matrix, with columns a1, n …, an, and if x is in , then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is, . Ax is defined only if the number of columns of A equals the number of entries in x. © 2012 Pearson Education, Inc. Slide 1.4- 12 MATRIX EQUATION Ax b Theorem 3: If A is an m n matrix, with columns m a1, …, an, and if b is in , then the matrix equation Ax b has the same solution set as the vector equation x1a1 x2a 2 ... xn an b , which, in turn, has the same solution set as the system of linear equations whose augmented matrix is a n b. a1 a 2 © 2012 Pearson Education, Inc. Slide 1.4- 13 EXISTENCE OF SOLUTIONS The equation Ax b has a solution if and only if b is a linear combination of the columns of A. Theorem 4: Let A be an m n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. m a. For each b in , the equation Ax b has a solution. b. Each b in m is a linear combination of the columns of A. m c. The columns of A span . d. A has a pivot position in every row. © 2012 Pearson Education, Inc. Slide 1.4- 14 SOLUTION SETS OF LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form Ax 0 , where A is an m n matrix and 0 is the zero vector in m . Nonhomogeneous systems Ax b © 2012 Pearson Education, Inc. Slide 1.5- 15 SOLUTION SETS OF LINEAR SYSTEMS When a nonhomogeneous linear system has many solutions, the general solution can be written in parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system. © 2012 Pearson Education, Inc. Slide 1.5- 16 SOLUTION SETS OF LINEAR SYSTEMS Theorem 6: Suppose the equation Ax b is consistent for some given b, 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. This theorem says that if Ax b has a solution, then the solution set is obtained by translating the solution set of Ax 0, using any particular solution p of Ax b for the translation. © 2012 Pearson Education, Inc. Slide 1.5- 17 1.7 LINEAR INDEPENDENCE Definition: An indexed set of vectors {v1, …, vp} in n is said to be linearly independent if the vector equation x v x v ... x v 0 1 1 2 2 p p has only the trivial solution. The set {v1, …, vp} is said to be linearly dependent if there exist weights c1, …, cp, not all zero, such that c1v1 c2 v2 ... c p v p 0 ----(1) Slide 1.7- 18 LINEAR INDEPENDENCE The columns of matrix A are linearly independent if and only if the equation Ax 0 has only the trivial solution. 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. Slide 1.7- 19 LINEAR INDEPENDENCE Theorem 7: Characterization of Linearly Dependent Sets An indexed set S {v1 ,..., v p } 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. Slide 1.7- 20 SETS OF TWO OR MORE VECTORS Theorem 8: If a set contains more vectors than there are entries in each vector, then the set is linearly n dependent. That is, any set {v1, …, vp} in is linearly dependent if p n . Theorem 9: If a set S {v1 ,..., v p } contains the zero vector, then the set is linearly dependent. Slide 1.7- 21 1.8 INTRODUCTION TO LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from » to » m is a rule that assigns to each vector x in » n a vector T (x) in » m. A transformation (or mapping) T is linear if: i. T (u v) T (u) T (v) for all u, v in the domain of T; ii. T (cu) cT (u) for all scalars c and all u in the domain of T. Slide 1.8- 22 n LINEAR TRANSFORMATIONS Linear transformations preserve the operations of vector addition and scalar multiplication. These two properties lead to the following useful facts. If T is a linear transformation, then T (0) 0 ----(3) T (cu dv) cT (u) dT (v) T (c1v1 ... c p v p ) c1T (v1 ) ... c pT (v p ) Slide 1.8- 23 1.9 The Matrix of A Linear Transformation Let T : » n -- » m be a linear transformation. Then there exists a unique matrix A such that T (x)=Ax. In fact, A=[T(e1), T(e2), ….., T(en)], where {e1, e2, ….., en} is the standard basis for »n . The matrix A is called the standard matrix for the linear transformation T. Slide 1.8- 24 The Matrix of A Linear Transformation A mapping T: Rn-Rm is said to be onto if each b in Rm is the image of at least one x in Rn. A mapping T: Rn-Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn. Slide 1.8- 25 The Matrix of A Linear Transformation Let T : Rn -- Rm be a linear transformation. Then T is 1-1 if and only if the equation T(x)=0 has only the trivial solution. Let T : Rn -- Rm be a linear transformation and let A be the standard matrix for T. 1) T is onto if and only if the columns of A span Rm. 2) T is 1-1if and only if the columns of A are linearly independent. Slide 1.8- 26