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M392 Lecture April 6th A linear expression in variables x,y,z is a sum of constants times those variables. Example: 3x + 2y + 7z is a linear expression. A linear equation sets a linear expression equal to a constant. Example: 3x + 2y + 7z is a linear equation. A system of linear equations is a list of equations that are to be solved simultaneously for the unknowns (variables) Example: Here’s a linear system of 3 equations in 3 variables 3x + 7y + 8z = 41 2x + y =4 4x +z=7 The simplest useful linear systems consist of 2 equations in 2 variables. For example, in school you used to solve equations by rewriting: 2x + 3y = 5 -----> times 3 ------>6x+9y = 15 3x + 4y = 7 -----> times 2 ------>6x +8y = 14 __________ Subtract to get y=1 Plug back into first eq:----->2x +3(1) = 5 ----> x = 1 Solution x = 1; y = 1 We say that the system has ONE solution. However, other systems behave differently The system below has NO SOLUTIONS. The equations contradict each other! 2x + 3y = 5 -------> times 2------>4x + 6y = 10 …….10 doesn’t equal 7!! 4x + 6y = 7------------------------->4x + 6y = 7 The system below has MANY SOLUTIONS. The equations repeat each other. 2x + 3y = 5 -------> --------------->2x + 3y = 5 4x + 6y = 10-----divide by 2----->2x + 3y = 5 In other words, this system of 2 equations in 2 unknowns is the same as the following very simple system: 1 equation, 1 unknown: 2x + 3y = 5 There are many ways to write down the (infinitely) many solutions. Here’s the one we use. Let y = s be any number. Then x = (5-3s)/2 For example, when s = 0, y = 0 and x = 5/2 When s =1, y = 1 and x = 1 ,,, and so on. Each of the three kinds of systems of 2 equations in 2 unknowns can be solved graphically, by sketching the graphs of the equations, which are straight lines in the x,y plane. Solutions are pairs (x,y) where the two lines meet. Case I. Two lines cross at a point: ONE SOLUTION Case II: Two lines are parallel: NO SOLUTIONS Case III: Two lines are identical: INFINITELY MANY SOLUTIONS Heuvers Chapter I: Matrices (plural of matrix) are used to simplify the bookkeeping when we solve systems of linear equations. An m x n matrix is a rectangular array of numbers with m rows, n columns. 3 4 5 and B = Example: A = 1 2 3 4 4 4 are 2 x 3 matrices. 3 3 3 1 2 C = 3 4 is a 3 x 2 matrix. 2 2 MATRIX ALGEBRA Multiply a matrix by a scalar ( a number) : Multiply every matrix entry by that number. 3 4 5 6 8 10 = 2A = 2 1 2 3 2 4 6 Add two matrices: Only possible when they have the same size For example, A and B are both size 2 x 3, and so they can be added 3 4 5 4 4 4 7 8 9 + = A + B = 1 2 3 3 3 3 4 5 6 Exercise: Find 2A + 3B Multiplying and dividing matrices are tricky operations. Be patient….. Matrix entries: we write For example aij for the entry in row i, column j of matrix A. a11 3; a23 3; c21 3; The transpose of matrix M is written M T , obtained by writing the rows of M as columns of a new matrix M T . In words: Row i , column j of M T is row j, column i of M T In symbols: ij ji . M M 7 4 7 8 9 T , then M = 8 5 Example: if M = 4 5 6 9 6 A row matrix is a matrix with one row. A column matrix is a matrix with one column. The transpose of a column matrix is a row matrix, and vice versa. If 1 C 3. 4 R T then C T 1 3 4 Recall that the dot product of ( 1 3 5) and (2 4 1 3 5 2 4 6 1 2 3 4 5 6 2 12 30 6) is Finally, we can define matrix multiplication. When you multiply A by B, the product is written AB For matrices, the product is defined only when The number of columns of A equals the number of rows of B In particular the product AA is defined only when A is square: same number of rows and columns. Note: you can’t multiply a 2 x 3 matrix by a 2 x 3 matrix. However, you can multiply a 2 x 3 matrix by a 3 x 2 matrix 3 4 5 B = Example: A = 1 2 3 1 2 3 4 Find the product M = AB 2 2 Multiplication Rule for matrices: If M = AB, then The entry in row i, column j of M is the dot product of Row i of A with the transpose of Column j of B For example row 1, column 1 of the product AB is ( 3 4 5) (1 3 2) = 3 1 + 4 3 + 5 2 = 3 + 12 + 10 = 25