Lecture 7: Introduction to linear systems Two pictures of linear systems Consider the following system of linear algebraic equations { x 2y =, 2x+y = 7. (.) Note that it is a linear system with two unknowns and two equations.. Row picture Draw the two straight lines l : x 2y = 0 and l 2 : 2x+y 7 = 0 in the XY-plane and the intersection (x,y) = (3,) is the solution..2 Column picture ( ) ( ) ( ) x 2y + = 2x y 7 ( ) ( ) ( ) 2 x +y = 2 7 (.2) (.3) x = 3,y =. (.4) ( ) It implies that the column vector b = is a linear combination of two column ( ) ( ) 7 2 vectors v = and v 2 2 =..3 The number of solutions. Only one solution. See (.). 2. Infinitely many solutions. e.g., { x 2y =, 2x 4y = 2. (.5) Copy right reserved by Yingwei Wang
3. No solution. e.g., { x 2y =, x 2y = 2. (.6) Note that case is called normal or nonsingular and cases 2 & 3 are called singular. See Figures 3..-3..3 in your textbook for the plots of the three cases. 2 Matrix forms of linear systems 2. Basics. Matrix: A rectangular array of numbers. An m-by-n matrix means it has m rows and n columns, denoted as A m n. 2. Column vector: a matrix with n rows and column. 3. Row vector: a matrix with row and n columns. 2.2 Matrix-vector multiplication. Simple case: 2-by-2: ( ) a a 2 a 2 a 22 2 2 ( x 2. General case: n-by-n: a a 2... a n a 2 a 22... a 2n...... a n a n2... a nn x 2 ) n n 2 ( ) a x = +a 2 x 2. (2.) a 2 x +a 22 x 2 2 x x 2. x n n n j= a jx j n j= = a 2jx j. n j= a njx j n. (2.2) 2 Copy right reserved by Yingwei Wang
2.3 Augmented matrix { x 2y =, 2x+y = 7. ( ( ) ( column picture 2 x +y = 2) 7) ( )( ( matrix-vector form 2 x = 2 y) 7) augmented matrix 2 2 7 (2.3) (2.4) (2.5) (2.6) Usually, the augmented matrix, coefficient matrix and right hand side vector are denoted as A #,A and b, respectively, i.e., A # = [ A b ]. 2.4 Special matrices. Identity matrix: I n =.... x 2 For any column vector x =, we have. x n x n n I n x = x. (2.7) 2. Exchange matrix: ( 0 0 )( x 3. Zero matrix: all of the entries are zeros. x 2 ) = ( x2 x ). 3 Copy right reserved by Yingwei Wang
3 Gaussian elimination 3. Motivation Questions: How to solve the linear system after we arrive at the matrix forms? Main idea: Two principles about Gaussian elimination: () to create zeros as many as possible in the coefficient matrix (augmented matrix); (2) do not make any change on the solutions. Definition 3.. The matrix E is called (Row-)Echelon Matrix: if it have the following two properties:. Every row of zeros lies below every row with a non-zero element; 2. In each row with a non-zero element, the first non-zero element lies strictly to the right of the first non-zero element in the proceeding row. Remark 3.. The above property says that if E has any all-zero rows, then they are grouped together at the bottom of the matrix. The first (from the left) nonzero element in each of the other rows is called leading entry. The above property 2 says that the leading entries form a descending staircase pattern from upper left to lower right. See the following examples. Several examples: 0 : Yes. (3.) 0 0 0 : Yes. (3.2) 0 0 0 0 0 0 : No. (3.3) 0 0 0 : No. (3.4) 0 4 Copy right reserved by Yingwei Wang
Definition 3.2. A matrix E is called (Row-)Reducd Echelon Matrix if it have the following two properties:. It belongs to echelon matrix; 2. Each leading entry is the only nonzero element in its column; 3. Each leading entry is one. Several examples: 0 0 0 0 : Yes (3.5) 0 0 0 3 : No (3.6) 0 0 0 0 2 R 2 +R R 0 3 : Yes. (3.7) 0 0 0 3.2 Elementary row operations The following are the three types of elementary row operations on the matrix A:. Interchange any two rows. E.g., R R 2. 2. Multiply any (single) row by a non-zero number. E.g., 2 R R. 3. Multiply any row by a non-zero number and add the result to any other row. E.g., 2R +R 2 R 2. Remark 3.2. Note that there is no change of the solutions to the original linear system after the elmentary row operations. Besides, those are the only three operations you are allowed to use in the Gaussian elimination. 5 Copy right reserved by Yingwei Wang
Definition 3.3. Two matrices are called row equivalent if one can be obtained from the other by a finite sequence of elementary row operations. Remark 3.3. If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solutions set. Example. Recall the linear system { x 2y =, 2x+y = 7. (3.8) can be rewritten as the following matrix form: 2. (3.9) 2 7 Starting from the matrix form (3.9), we can perform the elementary row operations: 2 2 7 ( 2) R +R 2 R 2 2 (3.0) 0 5 5 5 R 2 R 2 2 (3.) 0 2R 2 +R R 0 3. (3.2) 0 It follows that the solution to the linear system is x = 3, y =. 3.3 A 3-by-3 example with unique solution Example 2. Solve the linear system 2x y +z = 5, x y z = 4, 2x+3y +z = 6. (3.3) 6 Copy right reserved by Yingwei Wang
Solution: Start from the augmented matrix and performing the elmentary row operations to get the echelon form. 2 5 4 (3.4) 2 3 6 4 R R 2 2 5 (3.5) 2 3 6 4 ( 2) R +R 2 R 2 0 3 3 (3.6) 2 3 6 4 2 R +R 3 R 3 0 3 3 (3.7) 0 2 4 ( ) R 2 +R 3 R 3 0 3 3 (3.8) 0 0 4 5 4 ( /4) R 3 R 3 0 3 3 (3.9) 0 0 5/4 Now we can perform the so-called back substitution:. The third row in (3.9) indicates that 2. The second row in (3.9) tells us 3. The first row in (3.9) gives us x 3 = 5/4. x 2 +3x 3 = 3, x 2 = 3 3x 3 = 3/4. x x 2 x 3 = 4, x = 4+x 2 +x 3 = 7/2. 7 Copy right reserved by Yingwei Wang
The unique solution is x 7/2 y = 3/4. (3.20) z 5/4 Remark 3.4. We observe that the first entry in the matrix (3.9) is 2. There is another way to make it into : 2 5 /2 /2 5/2 4 (/2) R R 4. 2 3 6 2 3 6 8 Copy right reserved by Yingwei Wang