I have been doing some self study in preparation for the upcoming term and I realized I have very little intuition for doing linear algebra on spaces of functions. Can you recommend any linear algebra text books that have decent sections/chapters on things like dual spaces, endomorphisms and homomorphisms between spaces of functions?

An example of a problem that tripped me up is from a text book on manifolds, and they are introducing the topic of vector bundles and one of the exercises in the textbook is to show the existence of an isomorphism between the space of all $\binom{k}{l+1}$-tensors and the space of all multilinear functions:

$$ \underset{l}{\underbrace{V^*\times\cdots\times V^*}}\times\underset{k}{\underbrace{V\times\cdots\times V}}\longrightarrow V $$

I realized I didn't have the intuition for understanding how the linear algebra concepts generalize to these more complicated spaces I am comfortable with to these function spaces. I guess really what I am looking for is a textbook for undergraduate linear algebra at a higher level of abstraction so the focus isnt on vector spaces over $\mathbb{R}$ or $\mathbb{C}$. Any recommended readings to bring me up to speed would be appreciated!


Sheldon Axler's Linear Algebra Done Right, in my opinion, does a good job covering dual spaces and linear maps between general finite dimensional vector spaces.



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