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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Prove that the canonical mapping from an infinite dimensional vector space to it's double dual is a one-to-one mapping. [closed]
What is the canonical correspondence from a vector space V to it's double dual $V^{**}.$ Prove that this correspondence is one-one.($V$ need not be finite dimensional)
I tried solving the problem in ...
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How do I find maximal quotients of subspaces of the vector spaces $V_i$?
Say I have linear maps $V_1 \xrightarrow{h} V_2 \xleftarrow{g} V_3 \xrightarrow{f} V_4$. Assume $V_i$ are finite dimensional.
(1) I want to find maximal subspaces $W_i$ of $V_i$ such that in the ...
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What is the dimension of the following vector space? [closed]
I want to find the dimension of quotient
$$
V = \mathbb{R}^{2}{\LARGE /}\left\{\begin{pmatrix} \phantom{-}x\\-x\end{pmatrix} : x \in \mathbb{R}\right\}.
$$
It feels ...
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In an infinite dimensional vector space, is it true that ''a linear transformation is isomorphism if and only if it sends basis to basis.''?
In a finite dimensional vector space, a linear transformation is an isomorphism if and only if it sends basis to basis. I wonder whether this result will also hold for infinite dimensional vector ...
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Spanning set of support functionals in dual space
I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries:
Let $X$ be a normed space and $X^*$ be the ...
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Visualizing Matrix Spaces
When thinking about vector spaces a common example is vectors in $\mathbb{R}^n$. It is also very common to visualize this space as a set of as arrows in some n-dimensional space, with the typical ...
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Representing the finite field as $\{i*g+j\}$ where $g$ is a generator
This question arose from my thoughts on why the size of a finite field is always a prime power like $p^n$.
First, $\Bbb Z/p\Bbb Z$ is a field, and $\Bbb Z/p\Bbb Z -\{0\}$ is a cyclic group under the ...
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Given $A, B, C, D$ in $Oxyz$ space, find $M \in CD$ such that $MA + MB$ is smallest. Why can't I use AM-GM to solve this?
In the $Oxyz$ space, consider four points $A(-1, 1, 6),$
$B(-3,-2,-4),$ $C(1,2,-1),$ $D(2,-2,0).$ Find $M \in CD$ such that $△MAB$ has the smallest perimeter.
As $AB$ is constant, the task is ...
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Proof: If $B = \{b_1, ..., b_n\}$ and $ W = \{w_1, ..., w_n \}$ are basis of a vectorspace, then so is $\{b_1, ..., b_{k-1}, w_{k}, ..., w_{n}\}$.
I have some questions regarding the soundness of my reasoning in the following proof:
Proposition: Let $V$ be a (finite) vector space over $F$ with the
bases $B=\{b_1,b_2,\ldots, b_n\},\ W = \{w_1, ...
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Why is positive definite defined this way?
Norm: A norm on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb R$ which assigns each vector $x$ its length $\|x\| \in \mathbb R,$ such that for all $\lambda \in R$ and $x, y \in V$ the ...
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"field" vs. "vector field" [duplicate]
Is the "field" in the "vector field" as the same "field" in algebra: as the commutative ring with the multiplicative inverse?
If yes, then the "vector field" ...
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How do I find subspaces and quotients of the vector spaces $V_i$ such that the maps become injective and surjective?
Say I have linear maps $V_1 \xrightarrow{g} V_2 \xrightarrow{f} V_3$. I want to find subspaces $W_i$ of $V_i$ such that I have linear maps $W_1 \xrightarrow{\bar{g}} W_2 \xrightarrow{\bar{f}} W_3$, ...
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Action of matrices on right vector spaces
Let $V$ be a right (finite-dimensional) vector space over the (not necessarily commutative) division ring $\ell$. If we represent linear maps as matrices, then how do these matrices act on the ...
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Manifolds and Euclidean Spaces
This might be a basic question, but it's been irking me for the past few days.
The common definition of a manifold is as a second-countable, Haussdorff topoplogical space which is locally homeomorphic ...
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Is there a geometric interpretation of symmetric powers of vectors?
Let $V$ be a finite-dimensional vector space, over the real numbers to fix ideas. If we pick an inner product then the classical image of a vector in $V$ is a directed arrow. If $v_1, \ldots, v_k$ are ...
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