l1 - Math 20F Linear Algebra In Linear Algebra we solve...

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Math 20F Linear Algebra In Linear Algebra we solve systems of linear equations but linear algebra also provides a frame-work in which to think about many problems. Lecture 1: 1.1 Linear systems of equations. A linear system of m equations in n unknowns is of the form: (1.1) a 11 x 1 + a 12 x 2 + ... + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ... + a 2 n x n = b 2 . . . a m 1 x 1 + a m 2 x 2 + ... + a mn x n = b m where the a ij ’s and b i ’s are given constants and x 1 ,x 2 ,...,x n are unknowns to be determined. It is called and m × n system . A solution to the system is an ordered n -tuple ( x 1 ,x 2 ,...,x n ) such that all the m equations are satisfied. The set of all solutions are called the solution set . Let us try to understand the geometric meaning of a general 2 × 2 systems : a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 The solutions to each of the equations form a line in the ( x 1 ,x 2 )-plane. ( x 1 ,x 2 ) is therefore a solution to the system if and only if it lies on both these lines. Ex 1 Find all solutions to the system n x 1 + x 2 = 3 2 x 1 - x 2 = 0 Sol The two lines in the plane intersect at the point (1 , 2), which is the only solution Ex 2 Find all solutions to the system n x 1 + x 2 = 3 2 x 1 + 2 x 2 = 0 Sol The lines are parallel so they don’t intersect. No solutions! Ex 3
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l1 - Math 20F Linear Algebra In Linear Algebra we solve...

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