6
votes
Accepted
Endomorphisms of a countably infinite vector space.
I assume you mean that $V$ is countable-dimensional. Using elements of $I$ is actually a red herring; when generating a two-sided ideal you get to multiply on the left and the right by elements of $A$ ...
- 409k
2
votes
Accepted
In the vector space $(\mathbb R^n,+,⋅)$ over $\mathbb R$, does a scalar $s \in\mathbb R$ have any number of dimensions?
Note that the word "dimension" is used in multiple ways in math, and with slightly different definitions in each of them.
And when you mention "vector space", there is a highly ...
- 9,198
2
votes
Vector with no zero entries in the column space over a finite field
The other answer is correct regarding the NP-completeness of Syndrome decoding, which is about finding a vector with a nonzero lower bound on the Hamming weight in the nullspace of an appropriate ...
- 8,698
1
vote
Accepted
(Linear Independence) Show that for two vectors $v,w \in \mathbb{R}^n $, the conditions (i), (ii), (iii) are equivalent
(i) $\implies (ii)$
if $w\neq 0$ and $v=\rho w$ since $\Bbb R$ is a field exist the inverse of $\rho$ namely $\frac{1}{\rho}$ but this lead to a contradiction since $\frac{1}{\rho} \in \Bbb R, w = \...
- 806
1
vote
Vector with no zero entries in the column space over a finite field
The paper S. C. Ntafos, S. L. Hakimi, On the Complexity of Some Coding Problems, IEEE Trans. Inform. Theory IT-27:6 (Nov. 1981), states that the problem of checking whether there is a binary vector $x$...
1
vote
Must linearly dependent vectors pass through the origin?
Based on your "vector-spaces" tag, I'm assuming all your vectors in question are really elements of a vector space, say, $\Bbb R^2$.
Recall that when two points live on a line $\ell$ passing ...
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