Ok, what do we mean by: ? Now, what do we mean by ? Of course, the answer is “it depends”. The most common use of the first “equation” is “find the real number such that that number added to 2 equals 1.” In the second case, the most common use is “find real numbers such that this equation is true for all real .
In short, we are using the equal sign very differently: in the first case we are using it as the equivalence relation in the field of real numbers. In the second case, we are really talking about vector space equivalence.
We see this multiple use in calculus all the time; for example but Of course, the first is the usual non-oriented integral that we talk about in calculus courses (absolute values of the Jacobians!) and the latter is the oriented integral that we use for 2-forms, which, when you think about it, is the logical extension of the usual calculus I definite integral.
There are certainly more examples.
What got me to thinking about this was an office hour encounter I had with a numerical methods student (a good student who is doing solid work in the course). We were talking about various methods of solving the matrix problem where is a column vector of variables and is the “answer” vector of numbers. We were discussing the number of operations (multiplications/divisions and additions/subtractions) required to obtain a solution if we had that where was a diagonal matrix with non-zero entries, are lower and upper triangular matrices (respectively) with 1’s on the diagonal.
She kept on being off by a peculiar factor on the multiplication count.
Eventually we figured out the problem. When we converted the matrix equations to equations, she was counting the matrix entry multiplied by the unsolved for variables as a multiplication. Why? Well, once we solved for the variable we then counted operations with it AFTER it had been “solved for”. Example: given we don’t count the “coefficient times the variable” as a multiplication. But once we solve and obtain we then count operations involving . (of course, the diagonal elements are non-zero).
It is clear why we do this: prior to being solved for, the variables are really storage locations, and we are interested in counting the numerical operations that can contribute to round off error. But when we think about it, we are actually distinguishing between several types of multiplications: matrix multiplication, scalar multiplication in a vector space between a vector and a scalar, and the scalar (numerical) multiplication.
However, explaining that in class might lead to confusion among the students; it is probably best to bring this up only if someone is confused about it.
The language of mathematics can be so subtle that sometimes it probably good pedagogy to speak a bit informally, at least to beginning students.