Here are some lecture notes for a short course (perhaps between four and six 90-minute lectures) covering a decent variety of introductory linear algebra results. In preparing for the lectures, I spent a some time waffling between two approaches. The first is a "proof-oriented" approach, in which each lecture only introduces a few key results or applications, but a sufficient amount of time is allocated to illustrate the key steps of their proofs in detail. The second approach is "result-oriented," and abandons the hope of introducing detailed proof techniques in class, and instead introduces a rather substantial amount of results in a narrative-like fashion. I elected to go with the latter, as the students in our lab (where the first instance of these lectures was carried out) already had some background with how to prove the results, and there really are a lot of interesting results that are best introduced with some context.
We start from the basic properties of linear spaces and linear maps, working our way up to the deeper and more interesting results for operators on Hilbert spaces. The materials here include both key definitions and results as well as supplementary questions and illustrations, etc. used in class.
The contents are broken into four major topics. The first two are rather fundamental and of an introductory nature. The latter two get into some moderately advanced material. On the whole, the lecture contents are elementary, but go well beyond a first undergraduate course in linear algebra.
The points marked (*) are exercises that should be doable using just these materials and the assumed background (their difficulty varies from trivial to moderately involved exercises). The (**) exercises are somewhat more involved, and often require additional helper-lemmas and such. The materials are written in a fairly concise manner, and this was done intentionally, for reasons both sufficient and necessary. Sufficiency comes from the fact that students should develop and ability to "read between the lines" in this kind of content as soon as possible. Conversely, the material is already quite densely packed, and jamming in a lot of excess exposition I feel reduces readability significantly. It is certainly possible that a few errors still linger in these notes.
I used several classic texts, some newer and easier to find than others, while preparing these materials. Some linear operator and matrix algebra theory, some analysis and such; the English references are given below.
Axler, S. (1997). Linear Algebra Done Right. Springer, 2nd edition.
Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press, 1st edition.
Luenberger, D. G. (1968). Optimization by Vector Space Methods. Wiley.
Magnus, J. R. and Neudecker, H. (1999). Matrix differential calculus with applications in statistics and econometrics. Wiley, 3rd edition.
Mendelson, B. (1990). Introduction to Topology. Dover, 3rd edition.
Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill, 3rd edition.
Spivak, M. (1965). Calculus on Manifolds. Addison-Wesley.
As for the Japanese materials, since I basically just translated my materials directly, I didn't use a lot of Japanese reference texts. The authoritative intro is the superb Senkei Daisu Nyumon by Masahiko Saito, which I used to keep my translations of technical terms consistent across the materials.