System of Linear Equations
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1 Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given by a 11 x 1 + +a 1n x n = b 1, a m1 x 1 + +a mn x n = b m Consistent: It has solutions (one or infinitely many) Inconsistent: It has no solutions Solutions to a system of linear equations can be obtained by: Substitution (Convenient for small systems) Slide 2 Elementary Row Operations (Convenient for large systems) It is not needed to write down the variable x 1,, x n while performing the EROs Only the coefficients of the system, and the right hand side is needed This is the reason to introduce the matrix notation
2 Math 20F Linear Algebra Lecture 2 2 Matrix Notation Slide 3 System of Equations: a 11 x 1 + +a 1n x n = b 1, a m1 x 1 + +a mn x n = b m, Augmented Matrix a 11 a 1n b 1, a m1 a mn b m, n columns {}} { a 11 a 1n m rows a m1 a mn m n matrix Elementary Row Operations (EROs) Add to one row a multiple of the other Slide 4 Interchange two rows Multiply a row by a nonzero constant EROs do not change the solutions of a linear system of equations EROs are performed until the matrix is in echelon form Echelon form: Solutions of the linear system can be easily read out
3 Math 20F Linear Algebra Lecture 2 3 Definition 2 The diagonal elements of a matrix a 11 a 1n a m1 a mn Slide 5 are given by a ii, for i from 1 to the minimum of m and n Examples: Only the diagonal elements are given a 11 a 22, a 11, a 22 a 33 a 11 a 22 Echelon Forms Slide 6 Echelon form: Upper triangular (Every element below the diagonal is zero) Reduced Echelon Form: A matrix in echelon form such that the first nonzero element in every row satisfies both, it is equal to 1, it is the only nonzero element in that column
4 Math 20F Linear Algebra Lecture 3 4 Existence and uniqueness A system of linear equations is inconsistent if and only if the echelon form of the augmented matrix has a row of the form Slide 7 [0,, 0 b 0] A consistent system of linear equations contains either, a unique solution, that is, no free variables; or infinitely many solutions, that is, at least one free variable Vectors in IR n Slide 8 Definition, Operations, Components Linear combinations Span
5 Math 20F Linear Algebra Lecture 3 5 Definition, Operations, Components Definition 3 A vector in IR n, n 1, is an oriented segment Operations: Slide 9 Addition, Difference: Parallelogram law Multiplication by a number: Stretching, compressing In components: v ± w = v 1 ± w 1 = v 1 ± w 1, av = av 1 v n w n v n ± w n av n Some properties of the addition and multiplication by a scalar: Slide 10 u + v = v + u, u + (v + w) = (u + v) + w, a(u + v) = au + av, (a + b)u = au + bu
6 Math 20F Linear Algebra Lecture 4 6 Definition 4 A vector w IR n is a linear combination of p 1 vectors v 1,, v p in IR n if there exist p numbers c 1,, c p such that w = c 1 v c p v p A system of linear equations can be written as a vector equation: Slide 11 a 11 x 1 + +a 1n x n = b 1, a m1 x 1 + +a mn x n = b m a 11 x a 1n x n = b 1 a m1 a mn b m a 1 x a n x n = b Span Definition 5 Given p vectors v 1,, v p in IR n, denote by Span{v 1,, v p } the set of all linear combinations of v 1,, v p Slide 12 Note: Span{v 1,, v p } IR n, If w Span{v 1,, v p }, then there exist numbers c 1,, c p such that w = c 1 v c p v p
7 Math 20F Linear Algebra Lecture 4 7 Matrices as linear functions Slide 13 Definition 6 A linear function y : IR n IR m is a function y(x) of the form y 1 a 11 x 1 + +a 1n x n + c 1 =, a m1 x 1 + +a mn x n + c m y m where y = y 1 IRm, x = x 1 y m x n IRn, and a ij, c i are constants, with i = 1,, m, and j = 1, n Slide 14 Introducing the vector c IR m, and the m n matrix A as follows, c 1 a 11 a 1n c =, A =, c m a m1 a mn then, the linear function y(x) can be written as, y = Ax + c (Compare it with the expression for a linear function y : IR IR, that is, y = ax + c)
8 Math 20F Linear Algebra Lecture 4 8 Slide 15 The product Ax is defined as follows: a 11 a 1n x 1 a 11 x 1 + +a 1n x n = a m1 a mn x n a m1 x 1 + +a mn x n Exercise: Show that this product satisfies the following properties: A(u + v) = Au + Av, A(cu) = cau Slide 16 Summary A system of linear equations y 1 a 11 x 1 + +a 1n x n = a m1 x 1 + +a mn x n y m, can be expressed as a linear function, or as a linear combination of the column vectors, respectively, where A = [a 1,, a n ] y = Ax, y = a 1 x a n x n,
9 Math 20F Linear Algebra Lecture 4 9 Theorem 1 Fix and m n matrix A = [a 1,, a n ], and a vector b IR m Then, Slide 17 b Span{a 1,, a n } there exist x 1,, x n, such that b = a 1 x a n x n, b = Ax, the echelon form of [A b] has NO row of the form [0 0 b 0]
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