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I am looking for a text on linear algebra which is entirely basis free, and makes heavy use of the exterior algebra.

For instance, let $V$ be an $n$-dimensional vector space over a field $k$. Determinant of a map $\phi : V \rightarrow V$ may be defined as the map $\Lambda^n V \stackrel{\Lambda^n (\phi)}{\rightarrow} \Lambda^n V$, which is a scalar after choosing any basis of the one dimensional space $\Lambda^n V$. I am looking for a textbook which covers linear algebra in terms such as this.

Ronald J. Zallman
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  • I think you are mixing some things up in the definition of your determinant there, i.e. I would like to see a proof that the determinant can be uniquely determined/defined by a map between two one dimensional spaces, in this case $\Lambda^n V$... – Dirk Sep 03 '18 at 09:23
  • Would this answer your question? https://math.stackexchange.com/questions/21614/is-there-a-definition-of-determinants-that-does-not-rely-on-how-they-are-calcula – Ronald J. Zallman Sep 03 '18 at 21:05

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This pdf file looks rather interesting, although it is probably just a university report. It was published by the author through lulu.com: buy the book.

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