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Results tagged with linear-algebra
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user 2757
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
46
votes
1
answer
6k
views
Determinant of a determinant
Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m \time …
- 23.8k
16
votes
Accepted
Spheres over rational numbers and other fields
Any orthonormal set extends to an orthonormal basis, over any field
of characteristic not $2$. This is a special case of Witt's theorem.
EDIT: In response to Vipul's comment: The proof of Witt's the …
- 23.8k
14
votes
Accepted
Waring's problem for matrices
The answer is YES if $n$ is even. But if $n$ is odd, then the answer is NO since $-I$ is not a sum of two squares.
See
Griffin and Krusemeyer, Matrices as sums of squares, Linear and Multilinear Al …
- 23.8k
11
votes
Accepted
Surjectivity of bilinear forms.
The answer to Question B is no, as I'll show below.
Let $U=V=\mathbf{Q}^3$ and $W=\mathbf{Q}^4$. Define
$$\beta((u_1,u_2,u_3),(v_1,v_2,v_3))=(u_1 v_1,u_2 v_2,u_3 v_3, (u_1+u_2+u_3)(v_1+v_2+v_3)).$$
…
- 23.8k