Re:Linear Algebra: What matrices are
by wetmore on Saturday August 29, 2015 @03:17PM
(#10141062)
Linear operators are useful in many situations because
(1) They are "nice" operators which carry properties most functions can only dream of having: f(v + w) = f(v) + f(w), and f(av) = af(v). From these properties you can develop a very rich theory (along with the properties of vectors). These are the sorts of properties that we want all of our functions to have when we are young and first learning algebra - how many prealgebra students wish they could simplify (a + b)^2 to a^2 + b^2?
(2) A linear approximation is the first useful approximation for most behaviours, and at a small enough scale almost anything looks linear. Once you've made this approximation you get to exploit the properties of (1)
(1) They are "nice" operators which carry properties most functions can only dream of having: f(v + w) = f(v) + f(w), and f(av) = af(v). From these properties you can develop a very rich theory (along with the properties of vectors). These are the sorts of properties that we want all of our functions to have when we are young and first learning algebra - how many prealgebra students wish they could simplify (a + b)^2 to a^2 + b^2?
(2) A linear approximation is the first useful approximation for most behaviours, and at a small enough scale almost anything looks linear. Once you've made this approximation you get to exploit the properties of (1)