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I am a pure math major. I am taking a one-semester course covering the following topics:

Inner Product Spaces,
Direct Sums of Subspaces,
Primary Decomposition Theorem,
Reduction - (triangular, Jordan, rational, classical),
Dual Spaces,
Orthogonal Direct Sums,
Quadratic Forms

Our lecturer is a number theorist so he will probably make some links with number theory. He does not recommend any textbooks for the course but I learn best alone out of books designed for self-study. Lecture theaters full of whispering and distractions are hell to me! I like books that are written in a personal, quirky style that prompt deep reflection and encourage independent thinking. I like colorful motivating material!! I dislike mention of any "real-world applications" (ie physics/engineering). I hate when writers leave out big chunks in proofs and say the missing steps are "trivial" when they are not. I like it when solutions are available to the exercises.

Thank you for your help :)

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    $\begingroup$ Seems like Halmos's book fits your description except that his book is not colorful and that a solution manual does not exist (if you find it let me know!). $\endgroup$ – ares Sep 11 at 15:18
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    $\begingroup$ Possibly relevant: High-level linear algebra book $\endgroup$ – Dave L. Renfro Sep 11 at 15:36
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While studying this course, I used to refer to this book by Pramod Kumar which proved to be very useful and which I think has a lot of topics that are in your syllabus covered.

Also, I would recommend you giving Steven Roman’s book a try.

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I like Axler's Linear Algebra done Right as a simple, but powerful introduction into theoretical aspects of linear algebra, and Kostrikin-Manin's book as a very efficient reference for a second read.

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