Either of those things can help me identify a linear equation, so why is it that we are stuck with A(cx) = cA(x) and A(x+y) = A(x) + A(y) as the definition?
I’m not sure what you are referring to here. They certainly cannot always (or even usually) identify a linear equation. Those 2 things are going to be anywhere between useless and actively counterproductive in the vast majority of situations where you deal with potentially linear operations.
Indeed, if A is an n × n matrix of rank anything other than n − 1, the solution space of Ax=0 is not going to be a straight line. It will be a subspace of size n—rank(A), which can be made up of a single point (if A is invertible), a plane, a hyperplane, the entire space, etc.
“A function or equation with a constant slope that draws a single straight line on a graph” only works if you have a function on the real line, which is often just… trivial to visualize, especially in comparison to situations where you have matrices (as in linear algebra). Or imagine you have an operation defined on the space of functions on an infinite set X, which takes two functions f and g and adds them pointwise. This is a linear operator that cannot be visualized in any (finite) number of dimensions.
Bear with me because I am not a math major, but I am pretty sure “a linear equation is an equation that draws a straight line when you graph it” is a good enough explanation for someone to understand the basic concept.
So this is not correct, due to the above, and an important part of introductory linear algebra courses at the undergraduate level is to take people away from the Calc 101-style “stare at the graph” thinking and to make them consider the operation itself.
An object (the operation) is not the same as its representation (the drawing of its graph), and this is a critical point to understand as soon as possible when dealing with anything math-related (or really, anything rationality-related, as Eliezer has written about in the Sequences many times). Even the graph itself, in mathematical thinking, is crucially not the same as an actual drawing (it’s just the set of (x, f(x)), where x is in the domain).
Thank you for taking the time to explain that. I never took linear algebra, only college algebra, trig, and calc 1, 2, and 3. In college algebra our professor had us adding, subtracting, multiplying, and dividing matrices and I don’t remember needing those formulas to determine they were linear, but it was a long time ago, so my memory could be wrong, or the prof just gave us linear ones and didn’t make us determine whether they were linear or not. I suspected there was a good chance that what I was saying was ignorant, but you never know until you put it out there and ask. I tried getting AI to explain it, but bots aren’t exactly math whizzes themselves either. Anyway, I now stand corrected.
Regarding the graph vs the equation, that sounds like you are saying I was guilty of reification, but aren’t they both just abstractions and not real objects? Perhaps your point is that the equation produces the graph, but not the other way around?
I’m not sure what you are referring to here. They certainly cannot always (or even usually) identify a linear equation. Those 2 things are going to be anywhere between useless and actively counterproductive in the vast majority of situations where you deal with potentially linear operations.
Indeed, if A is an n × n matrix of rank anything other than n − 1, the solution space of Ax=0 is not going to be a straight line. It will be a subspace of size n—rank(A), which can be made up of a single point (if A is invertible), a plane, a hyperplane, the entire space, etc.
“A function or equation with a constant slope that draws a single straight line on a graph” only works if you have a function on the real line, which is often just… trivial to visualize, especially in comparison to situations where you have matrices (as in linear algebra). Or imagine you have an operation defined on the space of functions on an infinite set X, which takes two functions f and g and adds them pointwise. This is a linear operator that cannot be visualized in any (finite) number of dimensions.
So this is not correct, due to the above, and an important part of introductory linear algebra courses at the undergraduate level is to take people away from the Calc 101-style “stare at the graph” thinking and to make them consider the operation itself.
An object (the operation) is not the same as its representation (the drawing of its graph), and this is a critical point to understand as soon as possible when dealing with anything math-related (or really, anything rationality-related, as Eliezer has written about in the Sequences many times). Even the graph itself, in mathematical thinking, is crucially not the same as an actual drawing (it’s just the set of (x, f(x)), where x is in the domain).
Thank you for taking the time to explain that. I never took linear algebra, only college algebra, trig, and calc 1, 2, and 3. In college algebra our professor had us adding, subtracting, multiplying, and dividing matrices and I don’t remember needing those formulas to determine they were linear, but it was a long time ago, so my memory could be wrong, or the prof just gave us linear ones and didn’t make us determine whether they were linear or not. I suspected there was a good chance that what I was saying was ignorant, but you never know until you put it out there and ask. I tried getting AI to explain it, but bots aren’t exactly math whizzes themselves either. Anyway, I now stand corrected.
Regarding the graph vs the equation, that sounds like you are saying I was guilty of reification, but aren’t they both just abstractions and not real objects? Perhaps your point is that the equation produces the graph, but not the other way around?