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I am an undergraduate student. I am planning to do some project work on linear algebra. Until now i have studied transformations,elementary canonical forms ,rational and jordan canonical forms and their applications,inner product spaces and little bit of spectral theory. Please suggest me some topics which has some link with the above mentioned topics on which i can do my project work.

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Linear algebra is a wonderful subject. Here are some suggestions:

1) You have done a bit of spectral theory. Eigenvalues and eigenvectors are a fascinating and broad realm, you could delve into it in a more deeper level.

2) Linear algebra concepts apply to linear operators such as derivatives and other differential operators. Maybe you could explore the relationship between these two worlds.

3) More down-to-earth, linear algebra has important applications in statistics (linear regression on a cloud of points) and in data analysis problems such as how to find an interpolating polynomial function on $n$ given points in $\mathbb{R}^2$.

4) Dynamical systems. Given a system of first-order (not necessarily linear) differential equations, you can tell a lot about it by studying the Jacobian of the system and its eigenvalues.

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