Words, Episode 3: Linear

One of the obstacles to learning in mathematics and physics is the fact that there are many closely related concepts, although logically distinct. Additionally, the same structures (logical or mathematical) occur over and over again in our mathematical models of the world. For both reasons, the same word is sometimes used to mean several different things, with the meaning “clear from context.”

People who have studied such fields for a long time tend to do this out of laziness, often in casual conversation, with clear understanding between them of what sense of the word they mean.

Unfortunately, when you are just learning mathematics and physics, you don’t have much context (at least the vast majority of us don’t), and so you tend to get confused by these situations. Until, of course, you learn more, and then eventually you understand enough to tease out the logical structure of the subject.

The danger for a teacher is that if we stay lazy about this while teaching (or writing textbooks), then we will leave our students confused. We will do a tremendous service for our students if we would be kind and draw attention to such occurrences, and carefully compare and contrast the several meanings of such words, highlighting which meaning is relevant to the current discussion.

For example, consider the word “linear,” which means many things depending on context:

  • linear function: A linear function of one variable is of the form $y = mx + b$. The graph of such a function is a line, from which the appelation linear arises. However, the graph of a linear function in two variables, such as $z = Ax + By$ is a plane, not a line. The graphs of linear functions in more than two variables can no longer be visualized by we humans (at least I have not heard of anyone who can), but by analagy their graphs are called hyperplanes.
  • linear equation: A linear equation has the form $mx + b = 0$, or can be placed in this form by simple addition or subtraction of terms. For example, $5x + 3 = 2x – 1.7$ is a linear equation. This usage of linear is consistent with the first one in the following sense: If each side of the previous equation is considered to represent a function (that is, the left side represents the function $y = 5x + 3$ and the right side reprsents the function $y = 2x – 1.75$), then the graph of each function is a line. Furthermore, the equation represents a possible intersection point of the lines; there will be an actual intersection point if the equation has a solution.
  • linear combination of vectors: A linear combination of two vectors ${\bf u}$ and ${\bf v}$ is a vector of the form $a{\bf u} + b{\bf v}$, where $a$ and $b$ are numbers (let’s take them to be real numbers for simplicity). If ${\bf u}$ and ${\bf v}$ do not lie on the same line, then the collection of all vectors of the form $a{\bf u} + b{\bf v}$ is the plane containing ${\bf u}$ and ${\bf v}$.
  • linear differential equation: Suppose that the functions $y_1$ and $y_2$ are solutions to a differential equation. Then if the differential equation is linear, all functions of the form $ay_1 + by_2$, where $a$ and $b$ are real numbers, are guaranteed to also be solutions to the differential equation. You can see the analogy with a linear combination of vectors. To determine all possible solutions of a linear differential equation, it suffices to determine a number of independent basic solutions. Once the basic solutions are in hand, the collection of linear combinations of the basic solutions represent all possible solutions.
  • linear physical theory: A linear physical theory is one for which the differential equations of the theory (which describe the relevant phenomena) are linear. In this case, once one determines several solutions, a linear combination of the solutions is also a solution. In the context of physics, a linear combination of solutions is called a superposition of the solutions. Examples of linear theories are classical electromagnetic theory (described by Maxwell’s equations), and quantum mechanics (described in the non-relativistic case by Schrödinger’s equation). Einstein’s theory of gravity is a nonlinear theory.
  • linear medium: Despite the fact that classical electromagnetism is fundamentally a linear theory, electromagnetic phenomena in certain substances (called non-linear media) are nonlinear. That is, if cause A (acting on its own) creates effect B, and cause C (acting on its own) creates effect D, then if causes A and C act together, the effect is NOT the combination of B and D. A discussion of this situation in optics is here. For linear media, there is a superposition principle for causes and effects.
  • linear electronic device: See here.
  • linear system: Several equations that must all be satisfied is called a system of equations. If each equation is linear, then the system of equations is called a linear system, or a system of linear equations. Examples of linear systems of algebraic equations can be found here; examples of linear systems of differential equations can be found here.
  • linear transformation: A linear transformation from a vector space to another vector space is one that satisfies a kind of superposition principle. That is, if $T$ is an operator with domain vector space $V$ and codomain vector space $W$, then $T$ is linear if and only if the following two properties are satisfied: $T({\bf u} + {\bf v}) = T({\bf u}) + T({\bf v})$ and $T(k{\bf u}) = kT({\bf u})$, for all vectors ${\bf u}$ and ${\bf v}$ in $V$ and all scalars $k$. Once a basis is selected for the vector spaces $V$ and $W$, the matrix elements of a linear transformation are linear functions of the corresponding coördinates of $V$, which makes a connection between these two usages of the term linear. Furthermore, the range of $T$ is a hyperplane that passes through the origin in $W$. For example, if $V = W = \mathbb{R}$, then the transformation $T$ is linear if and only if the graph of $T$ is a line through the origin (or just the origin itself). This means that if the graph were a line that does not pass through the origin, the transformation is NOT linear, which is potentially confusing. This is an example of the many uses of the term linear sometimes dovetailing, sometimes clashing, in one example.
  • linear thinking: This is commonly used to mean the types of logical, step-by-step thinking that are important in constructing and verifying mathematical or scientific arguments, constructing algorithms and computer programs, and so on. This is contrasted with lateral thinking, divergent thinking, thinking outside the box, and so on, which are associated with creativity. Note that both kinds of thinking are essential for the creative, inventive, generative person.
  • linear media: For a discussion of marketing in the new nonlinear information world, see here.

We’ll leave bilinearity and multilinearity in mathematics for another post.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)