# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

87,976 questions
Filter by
Sorted by
Tagged with
490 views

### is there a generalization of unimodular matrices for non-square matrices?

Is there a generalization of unimodular matrices for non-square matrices? It is well-known that unimodular matrices guarantee an integral solution for a linear program (if the constraint matrix is ...
251 views

### figuring things out in linear algebra

There are things in linear algebra that I would like to better understand, from an intuitive point of view. For example, matrices are entities that may be use to transform a domain into another, by ...
21k views

2k views

173 views

1k views

213 views

### image of symmetric matrices under representation of $GL_2(\mathbb{R})$

Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...
196 views

### Characteristic polynomial and $p$-adic valuation

Suppose I had a linear operator $L$ whose characteristic polynomial was $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n-1}x + a_{n}$. Furthermore, I also know that the eigenvalues of $L$ have $p$-adic ...
94 views

### Condition to be able to decompose a finite-dimensional real vector space V into kernel and image of a linear map T from V to V

(I will phrase the question in terms of $\mathbb{R}^n$) Is the following statement a standard well-known linear algebra fact that I can quote without proving? (Perhaps more importantly, is it even ...
393 views