Basis change may be difficult.

Another topic which is quite nontrivial already at the technical level is the Jordan normal form of the matrix. Here perhaps one could show the students how this can be applied for solving of systems of linear first-order ODEs to motivate the importance of the topic.

@Robert Low, I completely agree with you, especially linear independence. That is even one that hits home for me when I was an undergraduate. Emphasis on it will be crucial for understanding what a basis is. It will help a lot of students if they choose to proceed into topics like Linear Programming where things like Basic Feasible Solutions are important.

Agree with the basis change, the eigen values/vector/basis etc problems ... assuming students are already dealing well with vectorial spaces (which imply to understand well notions of rank, span, basis etc.) ... Actually the basis of linear algebra.

As mentioned above, the concept of linear dependence/independence seems to be one of the more baffling concepts for students. This is an important aspect of dimensionless analysis in physics/engineering (upper-level undergrad courses) and every semester I wind up re-teaching these as part of the larger dimensionless analysis lectures.

After teaching a lot of linear algebra courses in the last ten years, I would say that most of the constructs you mentioned (linear independence, eigenvectors, abstract vector space) indeed pose problems in the beginning. However, I felt that the first huge obstacle, where most students stumble and some fail altogether, is the notion of dual space, the dual mapping, and the whole concept of duality.

Thank you to those that responded to the two or three most challenging topics in Linear Algebra. I agree with many that are mentioned. I was thinking the concept of Span, Eigen vectors, and Vector Space along with subspace might be candidates for student difficulties.

Dear @Robert,

I think that the most challenging issues for students in Linear Algebra are:

Transformation between bases;

Jordan Normal Form;

and the concept of linear independent.

Well, I believe that it depends on the scope of the course. If the course cover until duality an quotient spaces, then, without a doubt the concept of dual and quotient space are the most challenging ones. If, like most of the undergraduate courses, it only covers until Jordan the Jordan form then, I believe, the most challenging issues are: the construction of an algebraic structure from a set axioms, the concept of linear independence and concept of isomorphic vectorial spaces (for a "rookie" student is very hard to believe that a set of matrix and a set of polynomials are the "same" that $R^n$)

Going back two my first year in linear algebra, the geometrical

applications provided the motivation to go through the pain

of definitions and theorems:

Describe a mirroring, a translation, a rotation in 2D, in 3D.

Concatenate these geometrical mappings. What is the result?

How can it be described mathematically. Compute the rotation axis.

Extension: Concatenate affine linear mappings.

Furthermore: Classification of quadratic forms using the principal axis analysis.

The classics: Intersection or non-intersection of lines and planes, distance

measurement.

Working in an engineering environment, I have the impression

that a good geometrical intuition also helps when it goes to more complex

topics like fluid mechanics, structural mechanics, multivariate statistics.

And there is a close link between the

knowlegde of linear algebra and a spatial sense (intuition for geometry?).

While many fine suggestions have already been mentioned, let me suggest that most students struggle with linear algebra long before they have to make sense of transformations of basis or the meaning of an abstract vector space.  Even in courses restricting attention to Euclidian n-space (R^n  if you prefer), students struggle with understanding that a matrix or vector is an algebraic object itself rather than simply a convenient object to hold a bunch of real numbers.  Students focus on developing results by emphasizing what happens in each entry rather than what properties the entire object has.  Thus, faced with a question about invertibility, students struggle with what is happening in individual entries, rather than on the algebraic consequences for the entire matrix. As an example, ask your students to show that if v is a nonzero vector then v/(length of v) is a unit vector. At the start of a course, you will almost universally see proofs where each entry of the vector is divided by a squareroot of a sum of squares, and then sums of the squares of these entries are squared, ...  Rare is the student who invokes ||cu||=|c| ||u|| as the starting point, leading to a proof in about ten more characters.  Another point of struggle for students is captured in a problem such as "Show that the product of two upper triangular matrices is upper triangular."  In a first course, few students have the ability to handle the abstractions required to work with matrices of arbitrary size.  A much better question would be to ask students to show that the product of two 3x3 (or 4x4) matrices is upper triangular.  The underlying message of how the zero and nonzero entries align is demonstrated in a 3x3 or 4x4 case, and students do not have to worry about how to address row and column indexing in the more abstract environment.  Finally, most students arrive in a first course in linear algebra with at best a tenuous understanding about examples as illustrations versus examples as counterexamples versus generic elements of sets.  Since many linear algebra proofs begin with selecting a generic element of a set, and then demonstrating that that element possess a certain property, it is important that students understand what a generic element of a set is. Often, a question about a span of the vectors x,y,z that needs an answer based on a generic element of the span leads to a linear sum such as x+y+z  or perhaps cx+cy+cz, rather than ax+by+cz, or worse, to an expression such as "x,y,z" or "cx,cy,cz". 

I hope that this helps you with your preparation.  Good luck!

Jeff

Instead of wondering what the difficulties might be, you could wonder what tools will become central to solving problems and how students will begin to organize their thoughts. An analogy might be made to some mechanics for whom a big adjustable wrench is the go-to tool. It would be hard to think through a problem situation without reference to using a wrench. Several theorems in linear algebra can be seen as easy consequences of a few facts about row reduction. E.g. a row operation can be accomplished as left multiplication by an elementary matrix and when you reduce a square matrix, you end up with either the identity matrix or a row of 0's.

Lay's book emphasizes pivot columns, those with a leading 1 after reducing. When I had to use that book (department decision) it turned out that students could think confidently with some ingenuity and sophistication by referring to that notion and a few others. 

Craig, exactly, row and column operations and their implementation using matrix multiplication are indeed basic techniques. They lead to triangulation, diagonalization etc.