# Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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### Difference between isometry and euclidean motion

I know that they both preserve distance however what is the difference between them? I think isometry is part of Linear Transformations and euclidean motion is not
1 vote
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### Zero element in Lie algebra of a Lie group

On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth ...
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### What conditions must we put on a group in order for it to be a vector space over some field?

The axioms specifying the addition operation of a vector space are precisely those defining an abelian group, so of course it needs to be abelian. That's not sufficient though, as per this quora post.....
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### How every subspace is the solution space of a homogeneous system of linear equations?

There is this line in book "Mathematics for Machine Learning": " Every subspace $U ⊆ (R^n , +, ·)$ is the solution space of a homogeneous system of linear equations Ax = 0 for x ∈ $R^n$ ...
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### Theorem 5, Section 2.3 of Hoffman’s Linear Algebra

If $W$ is a subspace of a finite dimensional vector space $V$, every linearly independent subset of $W$ is finite and is part of a (finite) basis for $W$. Rephrasing Theorem to my taste: If $W\leq V$...
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### Can vector spaces occupy a large cardinal amount of dimensions?

So far I have found that a vector space of uncountably infinite dimensions is mathematically valid, but what about vector spaces that can occupy, say, an inaccessible or Mahlo cardinal amount of ...
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### Properties of homomorphism [closed]

Given f is an homomorphism from U to V, where U and V are two vector spaces, how can I prove the following propertis: If W is a third vector space, and g is an homomorphism from V to W, then is g ◦ f ...
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### Finding a Curve in a Vector Field that has 0 Flow

I am going through Vector Calculus for the first time and I had a thought which I assume has a neat solution, but I could not find a good answer for it: Imagine I have given 2D vector field, I want to ... 22 views

### Relation between subspaces and polynomials

Let $E = \mathbb{R}_{2022}[x]$. If F and G are in E, and $r + s \leq 2020$ where r and s are natural numbers and $$F=\{P(x) \in E: (x-2)^r | P(x)\}$$ and $$G=\{P(x) \in E: (x-22)^s | P(x)\}$$ find: a)...
1 vote
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### How do you call it when you stack vectors on top of each other?

How do you call it when you stack two vectors, let's say $u=\pmatrix{u_1\\u_2}$, $v=\pmatrix{v_1\\v_2}$, on top of each other such that you get $$u\oplus v=\pmatrix{u_1\\u_2\\v_1\\v_2}?$$ I found this ...
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### Theorem 4, Section 2.3 of Hoffman’s Linear Algebra

Let $V$ be a vector space which is spanned by a finite set of vectors $\beta_1,…,\beta_m$. Then any independent set of vectors in $V$ is finite and contains no more than $m$ elements. Question: (1) ...
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