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lecture_13 - MA 35100 LECTURE NOTES MONDAY FEBRUARY 15...

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MA 35100 LECTURE NOTES: MONDAY, FEBRUARY 15 Inverse of a Linear Transformation Consider an n × m matrix A , and the linear system A ~x = ~ y . Explicitly, a 11 x 1 + a 12 x 2 + · · · + a 1 m x m = y 1 a 21 x 1 + a 22 x 2 + · · · + a 2 m x m = y 2 . . . . . . . . . = . . . a n 1 x 1 + a n 2 x 2 + · · · + a nm x m = y n Say that we can add and subtract suitable multiples of the equations in order to arrive at a system in the form x 1 = b 11 y 1 + b 12 y 2 + · · · + b 1 n y n x 2 = b 21 y 1 + b 22 y 2 + · · · + b 2 n y n . . . = . . . . . . . . . x m = b m 1 y 1 + b m 2 y 2 + · · · + b mn y n This can be represented using matrices as ~x = B ~ y for some m × n matrix B . If it is possible to invert the system in this way, we say A is invertible, and denote B = A - 1 as its inverse. We review a familiar definition which discusses a general phenomenon. Definition. A function f : X Y is said to be invertible if for each y Y there exists a unique x X such that f ( x ) = y . If a function is invertible, we denote f - 1 : Y X as that function which sends y = f ( x ) to the unique x . Note that in general for an inverse to exist we need two statements: The equation f ( x ) = y has at least one solution, and the equation f ( x ) = y has at most one solution. We discuss a couple of examples. Consider the functions R R defined by f ( x ) = x and g ( x ) = x 2 .
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