All Questions
Tagged with cardinals linear-algebra
55
questions
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71
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An infinite linear system of equations with an uncountable number $A$ of equations
I will start with an example to make things clear and avoid confusion :
Take all $x>0$ and
$$\exp(x) = \sum_{-1<n} a_n x^n$$
Now finding $a_n$ can be described as an infinite linear system of ...
- 16.4k
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96
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Does $\dim \mathcal{L}(V,W) = \dim V \dim W$ hold for infinite dimensional vector spaces? If not when does it not hold? [duplicate]
I’m currently reading Alexer’s linear algebra done right. He proved in the book that $\dim \mathcal{L}(V,W) = \dim V \dim W$ holds if both $V$ and $W$ have finite dimension. I’m wondering if this ...
- 175
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1
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119
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Dimension of function space $X \to \mathbb F$
Let $X$ be an arbitrary set, and let $\mathbb F$ be an arbitrary field. Functions $X \to \mathbb F$ form a vector space over $\mathbb F$, with pointwise addition and scalar multiplication. What is its ...
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423
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How many 2D objects fit into a 3D object?
Hoe many times can you stack 2D objects before it becomes 3D?
I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional ...
- 11
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2
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315
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Is the rational span countable?
Let $\{x_n\}_{n\in\mathbb{N}}$ be a subset of an arbitrary vector space. Is the $span_\mathbb{Q} \{x_n\}_{n\in\mathbb{N}}$ countable?
I know that unions over countable index sets are countable and ...
- 110
2
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1
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230
views
Cardinality of vector space with uncountable basis
Let $V$ be a vector space over $\mathbb{R}$ with basis $\mathcal{F}, $ such that $|\mathcal{F}| = 2^{\aleph_0}$. Prove that $|V| = 2^{\aleph_0}.$
Clearly, $|\mathbb{R}| \leq |V|$ as the map $k\mapsto ...
- 1,225
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66
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Can a vector space with infinite basis be represented as a countable union of proper subspaces? [duplicate]
I am familiar with the case where the basis is not infinite (it becomes the case that it CANNOT be represented as such). However, many sources suggest that this is not the case for an infinite basis. ...
1
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1
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62
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Proof that vector spaces $k^M$ and $k^{(M)}$ are not isomorphic
Assume $k$ is a division ring and $M$ an infinite set. There are two vector spaces over $k$ to consider: $k^M$ and $k^{(M)}$ (the latter is the subset of $k^M$ containing all infinite sequence with ...
- 265
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votes
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332
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dimension of infinite dimensional vector space
Let $V$ be a vector space over a field $\mathbb{K}$ (either $\mathbb{C}$ or $\mathbb{R}$) that has an infinite linearly independent subset. Prove that if $B$ and $B'$ are two bases for $V,$ then $B$ ...
- 1,137
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Is a basis for the vector space of all series in $\mathbb{R}$ constructible [duplicate]
Given the $\mathbb{R}$-vectorspace $V =\mathbb{R}^\mathbb{N}$ of real valued series I was wondering if we can construct a Hamel-basis of $V$.
First of all I think that the dimension of $V$ is $ \mid\...
- 313
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1
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446
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What is the dimension of the vector space consisting of all real-valued functions?
The dimension of this vector space is obviously infinite dimensional, and it's not too much work to show that its basis is an uncountable set, making it an uncountably-infinite dimensional vector ...
- 31
1
vote
1
answer
56
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Infinite number of variables and infinite number of equations
Consider I have a system of linear equations $Ax=b$, where $A$ is a (countable) infinite by infinite matrix, $x$ and $b$ are infinite by 1 vector. If $A$ has infinite many zero rows, can we say the ...
- 1,249
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155
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Existence of a “direct union of subsets” operation (matching direct sum of vector subspaces)?
Dimension of vector space and cardinal number of sets have very similar properties :
Let $U$ be a vector space and $V, W \subseteq U$ two vector subspace, so
$$
\dim(V + W) = \dim(V) + \dim(W) - \dim(...
- 103
1
vote
1
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288
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Prove that any 2 bases of a vector space has the same cardinality
I know this question has been asked before, but I tried to prove it myself and I cant finish my prove because im not sure how to write the contradiction in a foraml and correct way.
Let V be a vector ...
- 3,751
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Explanation of Cardinal Arithmetic Used in Proving that bases of Vector spaces have the same cardinality.
Let $V$ be a vector space with bases $B_1$, $B_2$. For all $b\in B_1$ there exists $U_b\subset B_2$ such that $U_b$ is finite and $b\in span(U_b)$. Hence, $V=span(B_1)=span(\cup_{b\in B_1}U_b)$. Since ...
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