# Tagged Questions

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

23 views

### Computation of a linear transformation of symmetric algebras

The following is a problem and my attempt at a solution. I would appreciate any further guidance on intuition or neat tricks involving this problem and related concepts. Also, where can I read more ...
37 views

26 views

### Condition for nullity of quadrilinear form

I have been told the following. Lemma Suppose $V$ is a vector space over a field $K$, and $T:V\times V\times V\times V\to K$ is a multilinear map with the following properties holding for all ...
45 views

### Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
48 views

129 views

29 views

47 views

### Dual space of exterior power and exterior power of dual space

Let $V$ be a finite-dimensional vector space. Is there an isomorphism between $\Lambda^k(V^\ast)$ and $\left(\Lambda^k(V)\right)^\ast$? I was able to prove this with the additional requirement of ...
61 views

55 views

49 views

### Differential and Rank of $XAX^{-1}-A$

I have a map: $F_{A} (X) :GL\left(2n,\mathbb{R}\right) \longrightarrow\mathbb{\mathfrak{M}_{\mathit{2n\times2n}}\left(\mathbb{R}\right)}$ such as \begin{eqnarray} & F_{A}(X) & =XAX^{-1}-A \...
22 views

### Trace form seen by tensorial product.

I'm studying tensor product by the book "Multilinear Algebra - W. H. Greub". On page $38$ I couldn't understand how he got into the equation $(1.41)$. Could anyone help me?
68 views

### Generalized “scalar product” based on multilinear form?

In an $\mathbb{R}$-vector space $V$, the scalar product is a paradigmatic example of a non-degenerate, symmetric, positive-definite bilinear form $\beta : V \times V \to \mathbb{R}$. I wonder if the ...
37 views

### Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
56 views

### $M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...