What should I know how to do before I learn Linear Algebra

Question

I'm using Linear Algebra fourth edition by Friedman and i wanted to know what i should be able to do so that the course will not be so difficult. What proof-writing techniques should i be able to do? Are there any other books that would supplement this one well?

Answer 1

For proof writing: Having a solid handle on the fundamental logical operations and their relationships: and, or, not, implication, and biconditional; in symbols, $\land, \lor, \lnot, \Rightarrow, \Leftrightarrow$. Also, being able to recognize their use in mathematical arguments, as well as techniques for proving statements of basic forms:

For instance, $A\Rightarrow B$, can be proven by assuming $A$ and showing $B$ (direct proof). But it can also be proven by assuming $\lnot B$ (not $B$, or negation of $B$), and proving $\lnot A$. This is because $A\Rightarrow B$ is equivalent to $\lnot B \Rightarrow \lnot A$. Such a proof is known as a proof by contraposition, in many cases, this is also a proof by contradiction, since often, $A$ is the hypothesis, or assumptions of the proposition to be proved, hence, assuming the negation of your conclusion and proving $\lnot A$ is producing a statement ($\lnot A$) which contradicts your hypothesis (a contradiction, the hypothesis cannot be both true and false at the same time). Hence you reject the assumption, $\lnot B$ and conclude that it's negation, $\lnot\lnot B=B$ is true.

Similarly, proving $A\Leftrightarrow B$ is equivalent to $(A\Rightarrow B)\land(B\Rightarrow A)$, which can be proven by assuming $A$ and proving $B$, then assuming $B$ and proving $A$. Alternatively, one or both of the above implications can be proven by contraposition instead of directly.

One also sometimes needs to prove an or statement, for instance $A\Rightarrow(B\lor C)$, where $A$ is our hypothesis. The usual strategy is to pick on the easier of $B$ or $C$ to show, then, in cases: if the other is true, than you are done ($B\lor C$ is true in that case), otherwise...it must be false, and you can use that additional assumption to prove the easier case. For instance, suppose $C$ is easier, than you can argue, if $B$ is true, you are done, otherwise, assume not $B$ and then use $A\land\lnot B$ to prove $C$ follows in that case.

Knowing mathematical induction is occasionally useful for Linear Algebra, but only for a handful of proofs. (It's a pretty simple to technique to apply if you understand how the logic of a proof by induction works to actually show the truth of the proposition for every $n\in\mathbb{N}$).

Other than proof writing, for learning Linear Algebra specifically (the above logic stuff is important to any sufficiently rigorous mathematics class), the subject of Linear Algebra is rather vast, and there's easily more material than a single course could cover. Most modern introductory courses introduce the basics of the matrix algebra side of linear algebra, but don't do much more with it than that, and instead start connecting it to the vector space side of linear algebra and deal more with the theory there (it is more general because the vector space side can handle infinitely many variables just fine, but that is not so for matrix algebra). For this reason, imo, a good complement to a solid intro to linear algebra is a good book that goes further into matrix algebra than a standard linear algebra course would (you could also follow both up with an advanced book on linear algebra which might also cover topics like spectral theory and multilinear maps, I had the good fortune of seeing much on the latter as an undergrad in my first senior level class).

Another good thing to do would be to look at other fields of math and their connections with linear algebra, as linear algebra has a well-deserved reputation for finding use is almost every other field of math. A couple notable examples I am familiar with: Linear Algebra is deeply connected with multivariable calculus, and likewise is pervasive in differential geometry, it shows up in graph theory via the adjacency matrix of a graph, and the resulting spectral theory is the beginnings of algebraic graph theory.