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RelativeResourceManager-2 - Linear systems Linear equation:...

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1 Systems of Linear Equations Introduction & Methods Linear systems Linear equation: 3x + 5y - 18z = 0 (Contrast this to 3x 2 + 5yz = 0, which is NOT linear ) Sets of equations: 5x 1 + 8x 2 - 170x 3 = 15 8x 1 -3x 2 + 18x 3 = 17 ……. FIND SOLUTIONS! Solution sets : case 1 Infinite number of solutions (consistent), with some free parameters Solve the single equation 3x = 5 y Lots of possibilities. E.g. take x = t to be anything, then y = 3t + 5 Solution set is the set of all points { (t, 3t + 5) } An infinite number of solutions, described by the free parameter t Solution sets : case 2 no solutions, an inconsistent set 2x + 3y = 4 4x + 6y = 10 No solution at all since left hand side of second equation is just twice the first one, but right hand side is not. (4 5, hence no solution and inconsistent) (But x + 3y = 4 2x + 3y 4 4x + 6y = 8 would be OK)
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2 Solution sets : case 3 Unique Solution, consistent, no free parameters 2x + 3y = 7 x - y = 1 Has a unique solution: x = 2, y = 1 We need ways to decide a priori which case we’re dealing with. How to do this? ) by brute force 1) by brute force 2) by graphing the straight lines 3) by row reduction 4) by learning about determinants and matrices Computers of course can help How to solve? : brute force or high school approach 2x + 3y = 7 x + y 5 4x + y = 5 Summary: Solution Types for Systems of 2 equations in 2 unknowns (lines) Alternative to High School method Define a matrix (a rectangular array of numbers) at represents the coefficients in a set of linear that represents the coefficients in a set of linear equations
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3 Example of Matrices For any set of equations associate the matrix and augmented matrix 2x - 5y + 4z = 6 2x – y + z = 14 matrix 4x - 16y + 5z = 12 ugmented matrix 25 4 6 4 21 1 41 6 5
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This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.

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RelativeResourceManager-2 - Linear systems Linear equation:...

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