Linear Algebra I Lecture 8

Transcription

1 Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019

2 Outline 1 2

3 Gauss-Jordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n ) = 0. =. f m (x 1, x 2,..., x n ) = m with f i (x 1, x 2,..., x n ) = a i1 x 1 + a i2 x a in x n b i, we solve it using as follows: 1 Choose one of the equations, say f 1 = 0, such that a Use f 1 = 0 to eliminate x 1 from the other equations: k (a k1 /a 11 ) 1. 3 Repeat this process for x 2,..., x n.

4 Examples of Gauss-Jordan Elimination For example, let us solve { x + 2y = 1 1 2x + 3y = 2 2 We eliminate x from equation 2 using equation 1 : { x + 2y = 1 1 2x + 3y = 2 2 { { x + 2y = 1 x = 1 y = 0 y = 0

5 Examples of Gauss-Jordan Elimination For example, let us solve x + y + z = 1 1 x + 2y + 3z = 1 2 x + 3y + 4z = 2 3 First we eliminate x from equations 2 and 3 using equation 1 : x + y + z = x + y + z = x + 2y + 3z = 1 2 y + 2z = 0 4 x + 3y + 4z = 2 3 2y + 3z = 1 5 Then we eliminate y from equations 1 and 5 using equation 4 : x + y + z = x z = 1 x = y + 2z = 0 4 y + 2z = 0 y = 2 2y + 3z = 1 5 z = 1 z = 1

6 Matrix A matrix a 11 a a 1n a A = 21 a a 2n a m1 a m2... a mn is a rectangular array of numbers arranged in rows and columns. The above matrix A is an m n matrix with m rows and n columns. The i-th row of A is [a i1 a i2... a in ] a 1j a and the j-th column of A is 2j.. The (i, j)-th entry of A is a ij. a mj

7 Matrices Associated to a SLE Given a SLE a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. =. a m1 x 1 + a m2 x a mn x n = b m we associate two matrices to it: a 11 a a 1n a 11 a a 1n b 1 a 21 a a 2n a..... and 21 a a 2n b a m1 a m2... a mn a m1 a m2... a mn b m }{{}} [ {{ ] } A A b called the coefficient matrix and the augmented matrix.

8 Elementary Row Operations When we add a multiple of one equation to another, we add a multiple of one row to another in the augmented matrix. Actually, when we apply one of the following three operations to a SLE: 1 exchange two equations 2 multiply one equation by a nonzero number 3 add a multiple of one equation to another we equivalently apply one of the following three operations to the corresponding augmented matrix: 1 exchange two rows 2 multiply one row by a nonzero number 3 add a multiple of one row to another called elementary row operations on a matrix.

9 Equivalence under Elementary Row Operations Theorem Given a system of equations f 1 (x 1, x 2,..., x n ) = 0 1 f 2 (x 1, x 2,..., x n ) = 0 2. =. f m (x 1, x 2,..., x n ) = 0. m if we 1 exchange two equations, 2 multiply one equation by a nonzero number, or 3 add a multiple of one equation to another, the solutions of the system remain the same.

10 Equivalence under Elementary Row Operations Proof. Suppose that we exchange the i-th and j-th equations. We want to show {(x 1, x 2,..., x n ) : f 1 = f 2 =... = f m = 0} = {(x 1, x 2,..., x n ) : f 1 =... = f i 1 = f j = f i+1 =... = f j 1 = f i = f j+1 =... = f m = 0} which is obvious.

11 Equivalence under Elementary Row Operations Proof. Suppose that we multiply the i-th equation by c 0. We want to show {(x 1, x 2,..., x n ) : f 1 = f 2 =... = f m = 0} = {(x 1, x 2,..., x n ) : f 1 =... = f i 1 = cf i = f i+1 =... = f m = 0}. It suffices to show {f i (x 1, x 2,..., x n ) = 0} = {cf i (x 1, x 2,..., x n ) = 0}. If f i (x 1, x 2,..., x n ) = 0, then cf i (x 1, x 2,..., x n ) = 0. If cf i (x 1, x 2,..., x n ) = 0, then f i (x 1, x 2,..., x n ) = 0 since c 0.

12 Equivalence under Elementary Row Operations Proof. Suppose that we add c times the i-th equation to the j-th equation. We want to show {(x 1, x 2,..., x n ) : f 1 = f 2 =... = f m = 0} = {(x 1, x 2,..., x n ) : f 1 =... = f j 1 = f j + cf i = f j+1 =... = f m = 0} It suffices to show {f i = f j = 0} = {f i = f j + cf i = 0}. If f i = f j = 0, then f j + cf i = 0. If f i = f j + cf i = 0, then f j = (f j + cf i ) cf i = 0.

13 Gauss-Jordan Elimination on Augmented Matrices In conclusion, when we apply a sequence of elementary row operations to the augmented matrix of a SLE, the solutions of the corresponding system remains the same. Gauss-Jordan. We apply a sequence of elementary row operations to reduce the augmented matrix of a SLE to a matrix satisfying: if the i-th row of this matrix is with c ij the (i, j)-th entry, then [ ] c ij... c st = 0 for s > i and t j If c ij 0, then c ij = 1. Such a matrix is called an echelon form of the original matrix.