Linear Algebra

Linear Algebra has been described as the interplay between algebraic manipulations and geometric interpretations. See Appendix A.

Linear Algebra is taught to students who have already learned Calculus. This isn’t to say that it is more difficult than Calculus, but rather, there’s a strategy of teaching its concepts to students who already have some math experience on their resume.

Curriculum

• Three Dimensional Coordinate Systems
  • Majority of work (if not all) is in Cartesian Coordinates {x,y,z}
  • Polar Coordinates (useful for some things in Physics)
• Vectors
  • Linearly Independent Equations
    • Determinants
    • Linearly Independent Vectors
  • Vector Addition
  • Scalar Multiplication of Vectors
• Matrices
  • Matrix Addition
  • Matrix Multiplication
• Vector Space
  • Fields
  • Axioms

Matrix Multiplication may provide your first encounter with an operation that isn’t Commutative. (commutative-adj, commutativity-noun)

Appendix A

We liked “algebraic manipulations and geometric interpretations” but it put two questions on the table: what is a manipulation? what is an interpretation?

We felt we could say “manipulation” is something you do when you keep making algebraic changes to an equation, in particular, when you prove an equation is true by changing it until you get something like “2=2”.

We still have a question on the table, “what are we doing when we interpret geometry?” Does it simply mean that we’re doing something that is mathematical, but it isn’t calculation?”

Cautionary note on possible confusion: you do calculations with numbers all the time in a high school Geometry class, but that’s because “stuff” from algebra sort of, well, intruded upon the geometry. We can think of several reasons to accept this intrusion, the first two reasons being 1)giving the student more content to learn in that year-long course, and 2)students are more confident of their mastery of a math topic if it involves working with numbers (more opportunities for positive feedback, “yes, you got the right answer’). The downside is, later, when asked to think about geometry without algebra, the question seems really weird and quite difficult.