Tricks Of The Trade In Undergraduate Mathematics-Operators
Understanding mathematical objects is paramount to problem solving because if you don't even know the players involved you can't understand what the problem is in the first place. In my last post we saw what mathematics objects are and we dealt with two examples, one trivial and another requiring a bit more effort . In this post we focus on what operators are.
We encountered a problem asking for the range of f(x) = (arcsinx)²+(arccosx)² and to solve this problem we had to develop a relationship between arcsinx and arccosx , namely arcsinx+arccosx = pi/2. Realising that this sum gives pi/2 isn't at all obvious , infact it looks unlikely. Should it be a function of x just as the summands are? And yet it isn't which is what makes mathematics beautiful . The sum of two functions result in a different kind of object , a number, and what makes that possible ? An operator!
Operators are 'functions' that transform objects to either different objects or other forms of the same object. (an example is the natural logarithm operator , which when acted on 2 gives approximately 0.693) or attimes the same object .
Often times I see students write things like tan = sin/cos which quite frankly makes no sense since the notion of division makes sense for objects rather than operators. A more accurate stastatement would be that tanx=sinx/cosx, which in English is saying:
taking the tangent operation an angle x is the same as applying the sine and cosine operators on x and dividing the former by the lalatter.
The tangent operator given above is unary in the sense that it acts on a single object (in this case a number). Some other operators such as linear transformations transform entire objects to different objects for example T:|R³-->P_2(|R) which transforms vectors of the form (a,b,c) to polynomial of degree less than or equal to 2,ie dx²+ex+f where a,b,c,d,e and f are real numbers.
Another type of operator is a binary operator, basically one which acts on two obobjectof a given set S to get another element in S. Notice hear the conditions are much stricter than unary operators which can give different objects totally. Common examples of binary operators(or operations) are +,-,× and a less familiar one may be matrix multiplication .
Operators As Objects
Sure it doesn't make sense to write tan= sin/cosx but then again remember that f(x) = tanx can be interpreted to be an object which is determined completely by what x is. But then again f(x) is also a map that take in inputs , x and spots put output, tanx, so it's a valid operator. So what gives? Well nothing! In higher level math, depending on the situation, familiar objects can be seen as operators like f(x)=tanx and also familiar operators can be seen as objects like D-operators used in solving Ordinary Differential Equations which you'll see in about your third year, it's a High level Calculus class.
Thanks for reading and happy maths!