# 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.

87,197 questions
Filter by
Sorted by
Tagged with
3 views

### Trying to understand Grobner basis

While studying Grobner basis, I realized that creating a basis from a given set of polynomials is not that hard, it is reduced to solving with Gauss Jordan a system of equations. What I don't ...
17 views

### Direct sum of two subspaces - how to show

I haven't found this exact question yet on this site: If $U$ and $W$ are subspaces of the inner product space $(\mathbb{R}, V, +, \langle .,. \rangle)$, and I have to show that $U \oplus W = V$, ...
15 views

20 views

14 views

### How to compute (unipotent) radicals

My question follows some previous one, essentially this one. I want to understand, given an algebraic group $G$ (say linear), how to compute its radical and unipotent radical. The (unipotent) radical ...
13 views

### If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$ then…

If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$. If $x$ is an eigenvalue of $B$ then $x$ is real and positive. (True/false)...
71 views

### reference request / study plan - to build a solid foundation in mathematics for research in ML and optimisation [closed]

reference request / study plan - to build a solid foundation in mathematics for research in ML and optimisation
32 views

### (A−λI)x=0 and x≠0 iff det(A−λI)=0: Why [[1,1],[1,1]][[2],[3]] = [[5],[5]] ≠ 0 when det([[1,1],[1,1]]) = 0?

As refered to Why non-trivial solution only if determinant is zero, I wonder why \begin{gather} \begin{bmatrix} 1 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} 2 \\3 \end{bmatrix} = \begin{...
26 views

### Construction of tensor algebra: extending the multiplication from being defined on pure tensors

T.S.Blyth in his book Module Theory: An Approach to Linear Algebra defined multiplication on the tensor algebra $\bigotimes M = \bigoplus_{n \in \mathbb{N}}\bigotimes^n M$ by first defining it for ...
64 views

### Existence of a symmetric matrix $X$ such that $XBX = A$

let $A,B$ be two positive semi-definite $n\times n$-matrices such that $$\mathrm{Range}(B^{1/2}AB^{1/2})=\mathrm{Range}(B)$$ and $$\mathrm{Rank}(A)=\mathrm{Rank}(B)=n-1$$ so is there a real ...
147 views