When to Learn Linear Algebra in Relation to Multivariable Calculus
Linear algebra is a fundamental branch of mathematics that focuses on vector spaces and linear mappings between such spaces. It is a distinct and abstract form of mathematics, quite different from either single-variable or multivariable (vector) calculus. As a pure abstract mathematics, it involves studying mathematical structures and their properties without any specific reference to concrete dimensions or geometric figures, unlike multivariable calculus which deals with functions of two or more variables and their interrelation.
Eric Platt's Perspective
In addition to the specialized nature of linear algebra, Eric Platt suggests that interrupting a sequence of mathematical courses mid-stream can be detrimental. He emphasizes that if you need to take a multivariable or vector calculus class, you should ideally take it immediately following the preceding course in your academic program. This approach maintains coherence and continuity in your learning process, which is crucial for understanding complex mathematical concepts.
Concurrent Learning is Recommended
Given the interdependencies and mutual benefits of these subjects, it is often recommended to take linear algebra and multivariable calculus concurrently. While neither subject is strictly dependent on the other—that is, you can learn either without the other—both subjects offer significant advantages when learned together.
By taking these courses simultaneously, you can enhance your comprehension of linear algebra by applying it to vector calculus problems. Conversely, your grasp of multivariable calculus can be deepened through the lens of linear algebra. For instance, the concept of limits in vector calculus can be better understood via linear functionals, and derivatives and partial derivatives can be appreciated more fully as linear operators. The notion of divergence is also an example of a linear operator. These connections highlight the underlying unity and interconnectedness of these mathematical disciplines.
Flexibility in Learning Order
If concurrent learning is not feasible due to scheduling constraints, it is generally advisable to learn vector calculus first. Linear algebra is more abstract, and gaining mathematical maturity in this area can be beneficial. It is important to remember, however, that the primary objective is to understand the material thoroughly. The key is not to become too concerned about the exact order of learning these subjects, but rather to focus on mastering the content.
Studying in a way that ensures you understand each concept clearly will ultimately lead to success, regardless of the order in which the subjects are encountered. Both linear algebra and multivariable calculus are essential for a wide range of applications in science, engineering, and data analytics, and a solid foundation in one complements your understanding of the other.