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# 1.1 - MATH 232 a b Course is about Linear Algebra Came...

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MATH 232 a. Course is about Linear Algebra b. Came about through study of equations which are linear c. Has developed into a framework and set of tools which are basic and useful (1) to all of mathematics and (2) in many applications 1

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1.1 Systems of Equations Some motivating examples: Generally, we consider equations involving vectors vectorx = x 1 . . . x p ∈R p where R = real numbers, but for now simply consider p = 1 and the scalar equation 2 x = 3 for x ∈R . Solve using the tools of arithmetic: If 2 x = 3 then 1 2 · 2 x = 1 2 · 3, so x = 3 2 This seems trivial, right? But what about 0 · x = 3 What can x be? 0 · x = 0 What can x be? Notice: We have examples where there is a unique solution , no solution , or infinitely many solutions SUMMARY: ax = b has unique sol’n ( x = b a if a negationslash = 0) no sol’n (if a = 0 , b negationslash = 0) infinitely many sol’ns (if a = 0 , b = 0) 2
A more complicated example: Now take two variables x and y (so vectorx = bracketleftBigg x y bracketrightBigg ) E 1 2 x + y = 1 E 2 3 x 4 y = 3 manipulate one linear system to an equivalent one, i.e., with the same solution set .

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1.1 - MATH 232 a b Course is about Linear Algebra Came...

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