# Key theorems in undergraduate linear algebra

I've been asked by an high school student what are the $5$ major theorems in Lang's Linear Algebra (and therefore, by extension, in an undergraduate linear algebra course). Firstly, I bluntly said that the question holds a great deal of subjectivity and that I couldn't answer it.

However, coming to think of it, I realized that the question is indeed somewhat well-posed. Indeed, it could be natural to ask which of the theorems presented in a course are most useful for the prosecution of the study; and it requires some maturity to pick the key ideas that are most relevant for a subject as a whole.

So, I would like to ask what are the $5$ theorems that best epitomize an undergraduate linear algebra course and communicate the key ideas of the subject.

It depends on your view of what is crucial to the topic. In no particular order, here's what I think is critical for a linear algebra course.

• Rank-nullity Theorem: Very useful to show isomorphism
• Cayley-Hamilton Theorem
• Jordan Normal Form
• The Gram-Schmidt process
• Dual Spaces
• I will replace Dual Spaces with the spectral theorem, but I think you have the gist of the subject. Dec 23, 2014 at 5:28
• This is not undergraduate linear algebra. Ok, maybe it is, I mean I wish, I try, I get in trouble if I did this in my "introductory course". That said, in terms of what is important for linear algebra (delete the undergraduate bit), I endorse your list over mine. After my list, you still need a second course to really learn the sort of abstract linear algebra which matters to higher math. Dec 24, 2014 at 4:13
• @James This was indeed all covered in my first year lin. alg. course. Jordan normal form was reffered to as "classification of endomorphisms" and as "the most important and beautiful result you'll learn in your first year here". The rest were all done too, except with no in-class proof of Cayley-Hamilton (just a reference to one). Jul 5, 2015 at 18:40

Let's see. If I had to pick, I'll keep it $\mathbb{R}$,

1. The linear correspondence theorem; any linear combination of columns present in $A$ is likewise found in the row reduced form of $A$. It follows that the solutions of $Av=b$ are easily read from $\text{rref}(A|b)$.
2. A square matrix $A$ is invertible iff $Av=0$ has only $v=0$ solution iff $Null(A)= \{0 \}$ iff $Col(A) =\mathbb{R}^n$ iff $\text{det}(A) \neq 0$ iff $A$ does not have zero as an eigenvalue iff $Av=b$ has a solution for all $b$ iff $Row(A) = \mathbb{R}^{1 \times n}$ iff $rref(A)=I$.
3. $L: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear transformation if and only if $L(v)=Av$ for $A \in \mathbb{R}^{ \ m \times n}$ and matrix multiplication is defined such that the matrix of the composite is the product of the composed transformations. Consequently matrix multiplication is associative, but not commutative, as it is essentially a covert notation for function composition (ok, I cheated, there are two or three here)
4. The dimension of a finite dimensional vector space is well-defined; if there exists a finite minimal spanning set then the number of vectors in that set is an invariant for a given vector space.
5. A matrix is diagonalizable iff it has an eigenbasis.(but, I'd really like to replace this with every matrix has a real Jordan form)

Here's my list, but I'm curious what others will say...

1. Properties of the extended Echelon form.
2. The Four subsets theorem.
3. det(AB)=det(A)det(B).
4. Every finite n dimensional complex vector space is isomorphic to C^n
5. Every linear transformation is equivalent to matrix multiplication.

In case the names for 1 and 2 aren't standard, see Theorems PEEF and FS here.

• I would define the determinant as the product of the eigenvalues, which would make 3 trivial. What definition would you use?
– user507
Dec 24, 2014 at 15:59
• @BenCrowell If that's how you define determinant, which I would think is atypical, how do you actually calculate it? Usually one finds eigenvalues via the characteristic polynomial, which is defined using the determinant... Jul 5, 2015 at 7:41
1. Understanding the connection between matrix multiplication and composition of linear maps.

2. The rank-nullity theorem

3. Complex linear maps always have eigenvectors. Maybe that we can always find a basis where the matrix is upper triangular?

4. The Gram-Schmidt algorithm/theorem

5. The real and complex spectral theorems.