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Dear zhaoliang, here is the answer (from Gerstenhaber's thesis) to question 3. a) The maximal dimension of a space of $n$ times $n$ nilpotent matrices is $\frac {n(n-1)}{2}$. b) The subspaces of tha …

Let $K$ be a field and $V$ a $K$-vector space of infinite dimension $\aleph$ (some infinite cardinal). Then the dual $V^*$ of $V$ has dimension $(Card K)^\aleph$ [which is much bigger than $\aleph$, …

Sergeib's question asks about vector spaces without a natural basis. Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default situ …

Every finite-dimensional vector space is isomorphic to its dual. However for an infinite-dimensional vector space $E$ over a field $K$ this is always false since its dual $E^\ast$ is a vector space o …

Here is an elementary motivation for interior products . Suppose $V$ is an $n$-dimensional vector space . To every non-zero vector $ v\in V$ you can associate the complex $$ 0\to V\to...\to \La …

Here are three excellent books. Tensor Spaces and Exterior Algebra by Takeo Yokonuma. Translations of Mathematical Monographs, volume 108, AMS 1992 You can browse it in Google books here Laurent Sc …

Dear Ila, the linear algebra result you mention is due to Dedekind-Weber and was published in Crelle's Journal dated 1882, in their article "Theorie der algebraischen Funktionen einer Veränderlichen …