How to define linear equations in an introductory linear algebra class

Question

I believe there is an issue of clarification with respect to the definition of linear equations in many linear algebra texts.

Here is a typical one ``A linear equation in the $n$ variables $x_1,x_2, ..., x_n$ is an equation that can be written in the form $a_1x_1+a_2x_2+ ... +a_nx_n=b$ where the coefficients $a_1,a_2,..., a_n$ and the constant term $b$ are constants.''

No problem with the definition so far except that the text proceeds to state that $x^2-y^3=5$ is not linear because "it contains powers". There is no further explanation. Of course a well-prepared student would say: $(x+y+5)^2-(x+y)^2=35$ appears to contain powers and is a linear equation, how do you know that $x^2-y^3=5$ cannot be simplified?

I want to say the issue is with "[it] can be written" in the definition. That some operational explanation should be added.

Here are my two questions:

(a) Which introductory textbooks go deeper in the issue?

(b) How do you approach the issue? (Again this is introductory linear algebra, students come in with just calculus I as prerequisite).

Answer 1

This is a good point. Most textbooks I have used begin with the definition you supply, although many of them would not make assertions about $x^3 - y^3 = 5$ that aren't proved, unless they made clear that a proof might come later, or made clear that they were just glossing over details to make the essential point that the world of linear equations is not the world of all multivariable equations.

Your example $(x+y+5)^2 - (x+y)^2 = 35$ makes a point that is very well taken (i.e., there is some reduction of powers that takes place that isn't visually apparent, but requires a moment or two of thought). Formally speaking, the assertion that something can or cannot be put in a certain form requires proof, and the definition would seem to require that, and formal discussion of these issues can be thorny and alienating for beginners. (For example, it isn't a priori clear that a polynomial of degree $n$ can't be identically equal, as a function, to a polynomial of lower degree, and I have never seen a calculus textbook take the small amount of time it would require to explain this, although they all use the result, at least implicitly. See my answer to https://math.stackexchange.com/a/98370/18076 for more on that.)

For beginners I think it's best just to tell people to focus on linear equations and not worry about proving that things are not linear. (One way of proving it later in your specific example would be to derive geometric properties of solution sets of linear equations and show that the solution set to $x^3 - y^3 = 5$ does not have these properties. Of course, that gets into finding solutions to a polynomial system, and doesn't handle the case of nonlinear systems whose solution sets do coincide with solution sets to systems of linear equations. You can see how this could go down a long and winding road.)

One simplification might be to qualify "can be written" by restricting what you get to use when you rewrite, e.g., addition, subtraction, and scalar multiplication only. And make clear that this is a pedagogical expedient. I think the focus of "can be written" is simply that everybody wants something like $y = 2x + 5$ to satisfy the definition even though the variables aren't all on one side.

And you're not going to lose a lot of generality there, because linear systems that somehow masquerade as non-linear systems aren't that common in practice. By the time students get to anything like that, they will be able to handle it.

Answer 2

"how do you know that $x^2 - y^3 = 5$ cannot be simplified"

For a finite system with finitely many variables, one can take derivatives of the left-hand side. Namely, hold $y$ constant, differentiate with respect to $x$ twice, and collect the like powers--this is now a little easier to do, and that's is the only merit of this method. If you don't get a zero, the polynomial is nonlinear. Similarly for $y$: hold $x$ constant, differentiate with respect to $y$ twice. Do we get a zero? If not, the polynomial is nonlinear. For the polynomial to be linear, it is necessary and sufficient that each of these second-derivative tests (for each variable involved) gives us zero.

Trying the above test on $(x+y+5)^2-(x+y)^2=35$, we see that the second derivative w.r.t. $x$ on the left gives us $0$, and the same for $y$.

Generally, though, the definition of a concept does not always give an immediate recipe for testing each object as to whether it satisfies the definition. If you've seen limits in calculus, you probably noticed that the definition of a limit gives no recipe for determining whether a limit exists, and if it does, what its value is.

The issue of "linear equations" gets a little easier to deal with once the concept of a linear transformation is introduced. Unfortunately, many courses try to teach "linear equations" before introducing linear transformations.

An equation is linear if it is (or can be) written in the form $$Ax = b,$$ where $A$ is a linear transformation, $x$ is a sought vector in the domain of $A$, and $b$ is the given vector in the range of $A$.

This is similar to what you quoted from your text. However, it makes it easier to see that the linearity of the equation can be tested by taking two solutions, $x_{1}, x_{2}$ and seeing whether their difference is a solution of the homogeneous version $$ Ax = \vec{0} $$ of the original equation (think of this difference as a finite difference that approximates the first of the aforementioned derivatives). So, we are a little better off. If $A$ is given in a complicated, but explicit form, we can always consider numerical tests of linearity. Plus, this latter, coordinate-free, definition echoes the Superposition Principle, which pervades physics and engineering.

Just out of curiosity, I looked into Kurosh's Higher Algebra (a classic in its own time and a champion of clarity). It jumps into systems of linear equations from the get-go and doesn't seem too concerned with polynomials that, upon collecting the like powers, turn out linear. I haven't seen any textbook on linear algebra sweat this too much; perhaps, experience shows that testing linearity is usually not that much of difficulty, probably because linearity is so scarce.

Answer 3

If you want to give a basic idea to the students, I suggest you telling them this definition:

"If a function in x and y donot contain any powers on x and y terms, then the function is a linear function."

Now, the answer to your 1st question:

One of the books I trust is Introduction to linear algebra by Gilbert Strang, because I have myself read from this book.

Now since you mention that students know calculus, I would approach the problem the following way:

"If the slope (obtained by differentiation of the equation with respect to x) is a constant, then the equation is a linear one"

For example, we want to know whether x³-y³=9 is linear or not, then we do the following calculation:$$d(x^3)/dx - d(y^3)/dx = 0$$ $$3x^2 - 3y^2(dy/dx) = 0$$ $$dy/dx = x^2/y^2$$ Which is not constant as it depends on x and y.

Thus, $x^3 - y^3 =9$ is not a linear function.

Point to remember:

$dy/dx$ is called the slope of an equation and is constant for all linear equations.