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Linear Transformations, and thier properties

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Martin, Burda

Tuesday 18012011, Lecture notes by Y. Burda 1 Linear Transformations In the previous two lectures we have found the abstractions of notions of numbers and vectors in R . The abstractions were elements of a eld and vectors in an abstract vector space respectively. In this lecture we are going to nd the right abstraction of the notion of a matrix. When we are solving a system of linear equations Ax = b we arent interested in the matrix A itself, we are only interested in what is the result of multiplying this matrix by a vector x. Lets formalize this situation a little bit: Given an m n matrix A let f : A Rn m be a function that takes in a vector x and outputs the result of multiplying it by matrix A: n fA(x) = Ax for all x R 1 2 x 1 2 x x+2y For instance if A = ( 3 4 then f y ) = 3 4)(y) = 3x+4y . Of course not every function from R to R m is of such form. For instance every function which is of the form A for some matrix A satises fA(x+y) = fA(x) + f Ay) for all vectors x,y R (indeed: A(x + y) = Ax + Ay). Similarly every such function satises fA(x) = f Ax) for all R and x R . It turns out that every function that satises these two properties is of the form f for some matrix A: A Theorem. Suppose f : R Rn m satises n f(x + y) = f(x) + f(y) for all x,y R n f(x) = f(x) for all R,x R Then there exists a matrix A such that f(x) = Ax for all x R . n Moreover, the matrix A is the matrix whose columns are f(e ),1..,f(e ),n 1 where e ,...,e are the standard basis vectors for R : e = . ,...,e = 1 n 1 . n 0 0 . . 1 1
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