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
17,565
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Is F a vector space over F?
I know $F^n$ is a vector space over F (where F is a field)
This statement is true even when n is 1 right?
My professor said otherwise and that got me to ask the question here, please do excuse the ...
- 113
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Need help with part b [closed]
b) Find a basis for orthogonal complement of S, S
⊥. What is the dimension of S
⊥?
0
votes
2
answers
38
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Proving Y is a subspace of X and determining the dimension of Y.
I need help with two things. First, please confirm whether my proof below for part (i) is valid. Secondly, I'm not able to convince myself that the dimension of the subspace $Y$ described below is $n-...
- 692
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0
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15
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Squeeze transformation in higher dimension vector spaces?
From what I have read, including on Wikipedia and a post here on MSE, squeeze transformations are typically defined only for two-dimensional vector spaces such as a Euclidean plane, where one basis ...
- 83
2
votes
3
answers
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If $\mathbb{V}$ is a vector space, is it always true that $\mathbb{V} = \text{span}(\mathbb{V})$?
This question have just came to my mind after starting to study some linear algebra. I tried to write the following proof:
Let $\mathbb{V}$ be a vector space.
($\subseteq$) Let $v \in \mathbb{V}$. We ...
- 21
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votes
2
answers
53
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Dual norm of integral vector norm
For $x\in\mathbb{C}^n$, consider the following norm:
$\|x\|_{B} := \int_{B}|\langle x,y\rangle| \ d\mu(y),$
where $B:=\{y\in\mathbb{C}^n \ | \ \|y\|_2 \leq 1\}$ is the euclidean-norm ($\|\circ\|_2$) ...
- 11
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0
answers
51
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Relation between $\mathbb{Q}$ and $\mathbb{Z}^2$
Is there a way of writing $\mathbb{Q}$ in terms of $\mathbb{Z}^2$ in a simple expression?
My idea is to construct $\mathbb{Q}$ a quotient set of $\mathbb{Z}^2$, where $\mathbb{Z}$ is the set of ...
- 2,993
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2
answers
72
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What would be dimension of w1 intersection w2?
Let $M \in M_{2\times2} (\mathbb{R})$ be the vector space of all $2 \times 2$ matrices over $\mathbb{R}$, and let
$W_1 = \begin{bmatrix} x & y \\ 0 & x \end{bmatrix} \mid x, y \in \mathbb{R} ∧ ...
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0
answers
10
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Is nonlinear transform of a vector space a connected set
Consider $b_i=(\theta_p-\theta_q)\sin((\theta_p-\theta_q)\alpha)$ where $p,q\in\{1,\cdots,w\}$ and $N=\frac{w(w-1)}{2}$. In order to guarantee the one-to-one correspondence between $(\theta,\alpha)$ ...
- 15
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1
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39
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Necessary and sufficient condition to write a quadratic form on a finite-dimensional real vector space as a product of two linear functionals
I have come across a tricky linear algebra problem. We want to prove that a quadratic form $q$ on a finite dimensional real vector space $V$ can be expressed as $q(v) = f_1(v)f_2(v) \iff r + |\sigma| \...
- 380
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1
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25
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How to compute solid angle of $m$ vectors in $n$ dimensional space?
In 2d the angle between $2$ vectors were simply through the computation of the inner or outer product, and the "maximum" angle between $m$ vectors could be simply extracted by the maximum(...
0
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0
answers
18
views
Inverse Jacobian vs Pullback
Part of inverse Function Theorem tells us that for
$f$: M $\rightarrow$ N, $J(f^{-1})( \textbf{y}) = [(Jf)(f^{-1}( \textbf{y}))]^{-1}$.
That is the Jacobian of the inverse function at $ \textbf{y}\in ...
- 171
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1
answer
19
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Calculating a weighted midpoint between two 3D (XYZ) magnitude direction vectors
I'm really struggling. I have two "3D direction/motion magnitude" vectors for a 3D game engine.
The vectors consist of XYZ 3D components in the range <...
- 103
3
votes
1
answer
44
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linear combination - finite, infinite countable, and continuous
I am a beginner student of functional analysis.
We learn that, if $X$ is a vector space over $\mathbb{F}$ of finite dimension, it means it can be generated from a finite base, $V \subset X$, which ...
- 33
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votes
1
answer
33
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Intersection of 3 planes with linearly independent normals
In Tom Apostol's Calculus, vol. $1$, exercise $13.17.16$ is:
Prove that three planes whose normals are linearly independent intersect in one and only one point.
We know that every $n$ linearly ...
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Finding the index of the quotient group of $\mathbb{Q}^2$
Consider the subgroup $V$ of $\mathbb{Q}^2$, generated by
$$v_1=\left(\frac{1}{2},\frac{1}{3}\right),v_2=\left(\frac{1}{4},\frac{1}{5}\right),$$
in other words $V=\{av_1+bv_2:a,b\in\mathbb{Z}\}$. Let $...
- 165
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1
answer
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Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are spanned by given vectors
I have got the following entrance exam question.
Find a linear transformation $ T: \mathbb{R^4} \to \mathbb{R^3}$ such that $\ker T$ and $\operatorname{Range}T$ are respectively spanned by $$\{(1,1,1,...
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Why can we use the identity matrix when defining the characteristic polynomial?
We begin with $$\lambda v = Tv$$ where $\lambda$ is an eigenvalue, $v$ is an eigenvector, and $T$ is the transformation in question.
We state $$\lambda v - Tv = 0$$ We then must state $$(\lambda I - T)...
- 290
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0
answers
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V a IPS of finite dimension, $\phi : V \to F$ is a linear functional. Let $B = \{v_1...v_n\}$ an orthonormal basis for V.
Prove Riesz unique representation theorom:
If $v \in V$ is a vector in $V \implies \exists ! u \in V$ that stasfies $\phi (v) = \;<v,u>$
$\mathbf {Hint}$: express $u$ as a linear combination of ...
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$W$ is a vector space of all real $n\times n$ matrices $X$ such that $AX=0$ where $A$ is a real $n\times n$ matrix of rank $r$. Find dimension of $W$.
Let $W$ be the vector space of all real $n\times n$ matrices $X$ such that $AX=0$ where $A$ is a real $n\times n$ matrix of rank $r$. Find the dimension of $W$.
The number of free choices for ...
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votes
1
answer
42
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Relation between the regular representation and the group algebra.
Consider a group $G$. If I have understood the concept correctly, the vector space
$$
V=\operatorname{span}\{(\mathbf{e}_g)_{g\in G}\}
$$
is the regular representation when we define the action of the ...
- 719
2
votes
1
answer
33
views
Deduce why an arbitrary vector is the sum of kernels of endomorphisms
I had the following question in one of my exams:
Let $E$ be a non-zero finite dimensional vector space and let $f \in \mathcal{L}(E)$ be an endomorphism such that $f^3 = f$. Prove that, for any $x \...
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2
votes
1
answer
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Prove two functions on an inner product space $S,T: V \to V$ are linear
So far I have reached the fact that is pretty obvious:
$$
\langle v,T^*(u) \rangle = \langle T(v),u \rangle = \langle v,S(u) \rangle = \langle S^*(v),u\rangle
$$
thus receiving $T^*(u) = S(u)$ and $T(...
- 151
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votes
0
answers
12
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Show that a closed convex non empty subset of a normed vector space is the intersection of all closed semi-spaces that contain it. [closed]
I'm given the following problem:
Let $(X,\lvert \lvert \cdot \rvert \rvert)$ a normed vector space and let $C \subset X$ a non empty closed convex subset. Show that $C$ is the intersection of all ...
0
votes
1
answer
64
views
How does function mapping between two spaces $f \colon \mathbb{R}^2 \xrightarrow[]{} \mathbb{R}^2$ work?
This question will probably sound silly. In the mapping $f \colon \mathbb{R}^2 \xrightarrow[]{} \mathbb{R}^2$, with the function of $f(x) = x$, how does the dimension of the input equal the dimension ...
2
votes
2
answers
120
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Index Notation - An Inconsistency with a deeper meaning?
Background
Suppose we have a linear transformation $T : V \to W$ where $V$ and $W$ are finite dimensional vector spaces with bases $(e_i)$ and $(f_i)$ respectively. Using index notation, it is ...
- 4,978
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1
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Projections on a Hilbert Space over a vector subspace
Im trying to prove the following stament "Let $X$ be a Hilbert space and $M \subseteq X$ a closed vector subspace. For $x \in X$, we have that $$\bar{x} = P_M(x) \iff \bar{x} \in M\ \wedge\ \...
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1
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42
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Can a vector have more components than the dimension(D)
I read in a book that a 2d vector has 2 components and a 3d vector has 3 components.I know that a 2d vector has 2 components but can it have 3 components or more and can a 3d vector have 4 or more ...
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1
vote
2
answers
84
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Can you describe vectors in $\mathbb{R}^3$ without a basis?
For example, you can describe a vector in $M_{m \times n}(\mathbb{F})$ in terms of a coordinate vector of the standard basis or as the vector (matrix) itself. However, for something like $\mathbb{R}^n$...
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votes
1
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40
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Usual operations in a vector space [closed]
The set $$W=\left\{\left(x,y,z\right) \in \mathbb{R}^3 | x+y+z=0 \text{ and } 2x+y=0\right\}$$ with the usual operations is a vector space?
What is "usual operations"?
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0
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14
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Implicit description of a vector subspace with a homogeneous system
I was given the following problem:
Let $S = \{v_1, \ldots, v_4\} \subset \mathbb{R}^4$ where
\begin{align*} v_1 = (-1, 0, 1, 2), ~ v_2 = (3, 4, -2, 5), ~ v_3 =
> (0, 4, 1, 11), v_4 =
> ...
- 2,716
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0
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Assume $A \oplus B \cong A \oplus C$. Is $B \cong C$? [duplicate]
Let $E$ be a vector space. Let $A,B,C$ be vector subspaces of $E$. Let $\oplus$ denotes the direct sum of vector spaces. Let $\cong$ denotes the vector space isomorphism. If $B \cong C$ then $A \oplus ...
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The natural action of $Sp(V)$ on $V$ is Hamiltonian with the co-moment map given by: $\tilde{\mu} : sp(V) \to C^\infty(V); A \mapsto \tilde{\mu}_A$,
Given a symplectic vector space $(V, \Omega)$, consider the Lie group $G := Sp(V)$, consisting of all symplectomorphisms $\phi: V → V$. Show that if $(·, ·)$ is an invariant inner product, then the ...
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If $E \subset F$ then $\dim(V/F) \le \dim (V/E)$
Two Hamel bases of the same vector space have the same cardinality, so we define the dimension of a vector space as the cardinality of one of its Hamel bases. I'm trying to verify that
Let $V$ be a ...
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How is this identity valid? [closed]
I was reading about k means clustering where I came across this identity from the wiki and so far I’ve struggled to prove it. TBH I don’t remember much from the maths I did in college, please help me ...
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Kernel and Image Space Question
Let $T: \mathbb R^2 \rightarrow \mathbb R^3$.
$$
T\begin{pmatrix}x\\y\end{pmatrix}
=
\begin{pmatrix}x+y \\ 0 \\ 2x+3y\end{pmatrix}
$$
Find the span and basis for (a) Kernel of $T$, (b) Image space of $...
- 1,129
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0
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Question about Span & Basis
What does {$(1,2,3),(-2,0,4),(-1,2,7)$} span and find its basis ?
I form a matrix and reduce to echelon form:
$$\begin{pmatrix}
1 & 2 & 3\\
0 & 1 & \frac{5}{2}\\
0 & 0 & 0\\
\...
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1
answer
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Question about inner product of functions from a group to a field.
In The Symmetric Group the author, Bruce Sagan, writes on page 35 that the definition of the inner product is
$$
\langle\chi,\psi\rangle=\frac{1}{|G|}\sum\limits_{g\in G}\chi(g)\psi(g^{-1}),
$$
where $...
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1
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Can a vector with unordered components exist?
It seems like in order for a vector addition to be commutative, it needs to be defined in a "regular" manner, i.e. by adding matching vector components (because then the commutativity of ...
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1
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How to show vectors span a set?
Say if I have $V=${$(1,0,1),(0,0,1),(0,1,0)$} , how can I show that $V$ spans $\mathbb R^3$?
I believe there is a theorem that if the dimension of the vector space is $n$, then $n$ linearly ...
- 1,129
0
votes
1
answer
19
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Obtain the form and range of a matrix from its null space and left null space
The matrix 𝐴 has 𝑁(𝐴) [ 1 0 −1], and 𝑁(𝐴𝑇) [1 1 1 1] and [ 1 1 −1 −1]. What is the form of matrix 𝐴 and its range?
I known that the null space is related to columns and left null space is ...
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1
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1
answer
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How to construct a matrix given column and null spaces?
The problem says: If possible, construct a matrix whose column space contains [1 1 0] and [0 1 1] and whose null space contains [1 0 1] and [0,0,1].
I know that the column space has to be part of the ...
- 25
1
vote
0
answers
31
views
Showing that the set of vectors $e_k-e_l$ spans $ V=\left\{ (a_1,...,a_n) \ | \ a_1+\dots + a_n=0 \right\}. $
As the title says, I want to show that the set of vectors $$\{ e_k-e_l \ | \ k, l \in \{1, \dots, n \} \}$$ spans the vector space given by
$$
V=\left\{ (a_1,...,a_n) \ | \ a_1+\dots + a_n=0 \right\}.
...
- 63
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1
answer
33
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Can Rings (with unity) over themselves always be viewed as free modules?
Given a field $K$, we know that $K$ can be viewed as a 1-dimensional vector space over itself. Can this be extended to modules? In other words if I were to take a ring $R$ with unity and considered ...
1
vote
1
answer
27
views
How are $F^n$ and $F^\infty$ special cases of $F^S$? [duplicate]
I'm teaching myself Linear Algebra using Axler's Linear Algebra Done Right book and I'm confused about some definitions mentioned in the book.
The book defines $F$ as $\mathbb{R}$ or $\mathbb{C}$ ...
0
votes
1
answer
51
views
What does an infinite dimensional vector space even mean? What would be some good intuitive examples? [duplicate]
I am looking for some examples other than from quantum mechanics, if possible. Simple examples.
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2
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40
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Three-dimensional space position vector problem
In a three-dimensional space there exists a point $A$ and it's position vector $a$ is expressed as
$$a=\overrightarrow{OA}=\begin{bmatrix}
a_{1} \\a_{2}
\\ a_{3}
\end{bmatrix}$$
There is also a ...
- 259
0
votes
1
answer
26
views
Show that we have an induced quadratic form on quotient vector space
There is a quadratic form $q$ on a finite dimensional real vector space with associated symmetric bilinear form $\phi$. There is a subspace $R \leq V$ such that $\phi(r,v) = 0 \forall r \in R$ and $\...
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1
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0
answers
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views
Given subspaces $V$ and $W$ of vector space $U$, how do we denote the space of vectors that are in $U$ but not in $V+W$?
Let $U$ be a vector space and $V$ and $W$ two subspaces. It can be shown that $V+W$ is a subspace and that
$$\text{dim } (V+W)=\text{dim } V + \text{dim }W -\text{dim }(V \cap W)\tag{1}$$
I'd like to ...
- 4,346
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0
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What definition do you give these algebraic structures of "unmapped" (sorry for unrigorous definition) functions?
I know a bit of abstract algebra and of things like group theory. However, I know little beyond the very basic examples of the field. Forgive me for the lack of rigor or precise definition in what I ...
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