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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10 views

Where does the dualization of maps sign come from in graded vector spaces?

In the book Rational Homotopy Theory of Félix, Halperin and Thomas they state the following. Given linear maps $f:V\rightarrow W$ and $g:W'\rightarrow V'$ between graded vector spaces then we have $\...
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1answer
43 views

Are perpendicular vectors always in different subspaces?

So I understand when two subspaces are considered perpendicular and what it means for vectors to be perpendicular/orthogonal. The question I have is, if two vectors are perpendicular, do they always ...
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43 views

What can you say about $T$ if dim$(V) =$ Rank$(T - \lambda I)$?

I stumbled across this condition and I wanted to know what you could say about this: Let $T:V \to V$ be a linear transformation, with $V$ having a finite dimension. What can you say about $T$ if ...
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1answer
16 views

Vector spaces over an integral domain and the canonical isomorphism between the tensor products

Let $A$ be an integral domain and write $S=A-\{0\}$. Then the total ring of fractions $S^{-1}A$ of $A$ is an abelian field. Note that $\varepsilon:A\rightarrow S^{-1}A,\,a\mapsto a/1$, is an injective ...
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2answers
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Finding Intersection of two subspaces

My attempt: After substituting variables I got basis of V = { (1,0,0,0),(0,-1,1,0),(0,-1,0,1)} and W = {(-1,1,0,0),(0,0,4,1). These are also the basis of these subspaces so dim(V) = 3 and dim(W) = 2 ...
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19 views

Can we recover the bases of two infinite-dimensional vector spaces into a tensor product?

I know that in general, if V,W are infinite-dimensional vector spaces with basis {${v_i}$} and {${w_j}$} respectively it holds that V⊗W has as basis {${v_i⊗w_j}$}. My question is: what about the ...
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Showing the properties of an isomorphism between finite dimensional vector spaces.

$ \phi: V \rightarrow W $ an isomorphism between finite-dimensional vector spaces. For a subspace $U$ of $V$ we can write $ \phi(U):= {\{ \phi(u): u \in U} \} = \operatorname{im}( \phi | _{U})$ ...
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$(X,\tau)$ topological vector space (Hausdorff) is: locally compact $\iff$ of finite dimension.

I am trying to show that in any finite dimensional normed vector space, the unit ball is compact. To do this, I first proved that: $(X,|\cdot |_X)$ normed vector space is: locally compact $\iff$ the ...
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1answer
19 views

Finding a basis vector from a vector image of a linear mapping.

Let $\Bbb Q^3$ and $v = (1, 2, 3 ) \in V. $ Find a basis $B$ for $V$, so that $\operatorname{Im} v$ wrt $B$ $(1,0,0)$ has i.e. $(v)_{B} = (1,0,0)$. Should I find in this exercise a transformation ...
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23 views

Not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.

I am reading some notes about tensor product of vector spaces (those in here) where the following sentence can be found: Note that not every matrix on V ⊗ W can be written as a tensor product ...
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1answer
23 views

Proving that the mapping is linear: $ ( . )_{Q} : V \rightarrow K^{n} $, $ v \mapsto (v)_{Q} $

The Problem: $V$ finite dimensional Vectorspace and let $Q=(v_{1},...,v_{n})$ a basis of $V$.Show that the mapping $ ( . )_{Q} : V \rightarrow K^{n} $, $ v \mapsto (v)_{Q} $ linear is. ...
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1answer
37 views

Equivalence for artinian and noetherian vector spaces

I'm trying to prove the next proposition: For a vector space $V$ over a filed $F$, the next are equivalent: a) $V$ has a finite dimension b) $V$ is a finitely generated module c) $V$ ...
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1answer
38 views

Square matrix with complex entries whose space of eigenvectors have dimension $1$

Let $A\in M(n,\mathbb C)$ be a matrix such that the space of eigenvectors have dimension $1$. Hence $A$ has exactly one eigenvalue (because distinct eigenvalues give rise to linearly independent ...
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2answers
69 views

$ker(T) + ker(S) = V \implies Im(T + S) = Im(T) + Im(S)$

Let $V$ be a vector space. Let $T, S$ be two linear operators $T:V \rightarrow V$, $S: V \rightarrow V$, such that $$ker(T) + ker(S) = V$$, then we must have $$Im(T+S) = Im(T) + Im(S)$$. If the ...
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1answer
32 views

Distance to the boundary

Let $F$ be a closed set in $\mathbb{R}^n$ and $\Omega:=\mathbb{R}^n\setminus F$. For $x\in\Omega$, do we have ${\rm dist}(x,F)={\rm dist}(x,\partial \Omega)$?
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Shadow rushed by a disk in space

A disk of radius 1 is centered at the point $A(0,1,2)$ and is parallel to the plane $xOy$. A source of light is placed at the point $P(0,1,4)$. Characterize analytically the shadow and the disk rushed ...
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Find the vector function $r(t)$ which satisfies $r'(t)= \langle sin9t, sin9t, 9t \rangle$ and $r(0)= \langle 5,3,2 \rangle$.

I'm not sure how to do this problem at all, but would it make sense to plug in $\langle 5,3,2 \rangle$ into $r'(t)$?
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Show that the uniformly continuous functions on 𝑋 build a function space

Show that the uniformly continuous functions on $X$ build a function space. My attempt: Let $\mathbb{F}(X)$ be the set of uniformly continuous functions on $X$ (1) We show $\forall g, f \in \mathbb{...
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1answer
21 views

Prove that (MB (φ))^k = MB (φ^k)

Let V be a K vector space with base B: = {b1,…, bn} and φ an endomorphism in V with a representation matrix MB(φ). Prove that (MB(φ))^k = MB(φ^k) for k = 1, ..., n applies.
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35 views

find the angle between the affine spans

Let $W$ be a vector space and $U,V \subseteq W$ be subspaces. Let $p, q \in W.$ The angle between the affine spaces $P = p+ U$ and $Q = q + V$ (here $p+U := \{p+u: u\in U\}$) is defined as $\theta(P,Q)...
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Prove W is a subspace of V. [duplicate]

If W₁ ⊆ W₂ ⊆ W₃......, where Wᵢ are the subspaces of a vector space V, and W = W₁ ∪ W₂ ∪...... Prove that W ≤ V. So I proved that: If W₁ and W₂ are two subspaces of V and W₁ ∪ W₂ ≤ V then W₁ ⊆ W₂ or ...
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1answer
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Linear functionals on $n \times n $matrices over $ \mathbb{F}$

I need to show that all linear transformations on the above vector space is of the form $f_B(A)=trace(B^TA)$ for some $n \times n$ matrix $B$.How to proceed?I have just been able to show that $f_B$ is ...
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1answer
36 views

Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$. Then show that Span of $(V/W)= V$.

Let $V$ be a finite dimensional vector space and $W$ is proper subspace of $V$. Then show that Span of $(V/W)= V$. I am trying to show that $V/W$ contain a basis of $V$ but How to proceed ? any ...
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2answers
40 views

$\dim V \geq \dim U$, then there exists a linear injection from $U$ to $V$.

Let $U$ be a $K$ vector space with $\dim U=n$. I have to show that for all $r\in \mathbb{N}$ with $n\leq r$, there is a $K$ vector space $V$ with $\dim V=r$ and a linear injection $U\...
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45 views

Is there a formula to extract the real part of a complex vector?

For a complex number $z\in \mathbb{C}$, the formulas for the real part and the imaginary part the well-known formula: $$ \Re[z]=\frac{z+\overline{z}}{2}\\ \Im[z]=\frac{z-\overline{z}}{2i} $$ What is ...
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33 views

Show: Linear mapping or none [closed]

Could someone explain me if the following is a linear mapping or none: $$\text{Of }f:\mathbb{R^3}\to\mathbb{R^3} \text{ is known: }$$ $$f(\begin{pmatrix}1\\2\\3 \end{pmatrix})= \begin{pmatrix} 5\\3\\1 ...
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1answer
26 views

Notation question: dual space basis

I have an exercise that I am trying to decipher, but as I have never seen this notation before I do not know how to read it. The problem states: The vectors $x_1=(1,1,1),x_2=(1,1,-1)$ and $x_3=(1,-...
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1answer
14 views

What is the dimension and base of the following vectors' sum and intersection?

I have 2 Vector Subspaces of $\mathbb{R}^3$, namely $U = \operatorname{Span}(\begin{pmatrix} 2 \\ 5 \\ 9\end{pmatrix}, \begin{pmatrix} 0 \\ -1 \\ -3\end{pmatrix})$ and $W = \operatorname{Span}(\begin{...
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1answer
21 views

Relation between Jordan Normal Form and cyclic modules

I've just started reading about the relation between cyclic modules and Jordan Normal Form and, being honest, I've quite a doubt. The text I am using says that "clearly", the following assumption is ...
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1answer
26 views

Unique differentiable linear operator mapping $\mathbb{R} \rightarrow \mathbb{R}^N$?

I am trying to find a mapping $\phi$ such that: $\phi$ uniquely maps $\mathbb{R}$ to the subspace of $\mathbb{R}^N$ (for bounded $N$), where every dimension of the vector is bounded between $-1$ and $...
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15 views

Are hypothesis spaces (in machine learning) implicitly always vector spaces?

Intuitively, if the hypothesis space is very restrictive, then it might not be a vector space. But I've seen papers putting hypothesis (functions) into matrices, so I assume whenever the hypothesis ...
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29 views

Set of the linearly independent solutions of a linear ODE

Is it true that for all linear ordinary differential equations (ODEs), one can always find a set of all linearly independent solutions form a complete set (in the sense that any arbitrary solution is ...
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3answers
86 views

Proving that removing any vector of the linearly dependent set gives a linearly independent set

Consider the matrix representing 6 linearly dependent vectors: $$\left(\begin{array}{llllll} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 ...
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1answer
66 views

Why is $1,a,a^2,…,a^{n-1}$ linearly independent?

I have a basic question about the proof of "Every finite field extension is algebraic". Given the extension $K\subset L$ with $n:=[L:K]$ and $a \in L$, the proof says, that we have a linearly ...
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1answer
19 views

Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$

Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$ How do you prove this using Holder's inequality?
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2answers
35 views

Show that there is a $k \in \mathbb{N}$, for which applies: $f^{k}(V)=f^{k+l}(V)$ for all $l \in \mathbb{N}.$

I have a problem with the following task: Let $V$ be finite-dimensional $\mathbb{K}$ -Vector space and $f: V \rightarrow V$ an endomorphism. We define $f^{0}(V)=V$ and $f^{i+1}(V)=f\left(f^{i}(V)\...
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1answer
24 views

How to find a basis for the vector space of polynomials s.t. $f(-1) = f(2)$ of deg equal or less than $2$?

I have the following vector space and I want to find a basis for it and after that complete it to be a basis of the entire vector space os polynomials of degree $2$ or less. I got the following ...
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1answer
27 views

If an affine subspace doesn't intersect an affine hyperplane $H$, is it parallel to a subspace of $H$?

Let $A$ be an affine subspace of $V$, say $A = x + U$ for some vector $x$ and some subspace $U$, and suppose $B$ is an affine hyperplane of $V$, say $B = y + H$ for some vector $y$ and some hyperplane ...
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2answers
23 views

Find a normal vector $n$ to the plane $z−8(x−6)=2(3−y)$.

To find the normal vector, I need $ax+by+cz=d$ but with this question I'm not sure what that would be.
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Find the vector form of the equation for the line of intersection of the planes $x+y+z=8$ and $x+z=0$ [closed]

So $n_1= \langle 1,1,1 \rangle$ and $n_2= \langle 1,0,1 \rangle$. If we cross multiply these we get $i-k$ or $\langle 1,0,-1 \rangle$. From here I set $y=0$ since the second plane has a $y$ value of $...
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24 views

Calculate coordinates of polygon's points after changing lenght of one side

I have a list of points (vectors) on the surface, together they form a polygon. Two sides lead from each point, each side has only two points, these sides form the polygon. And my problem is: How to ...
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5answers
41 views

Find points where the plane $6x+y+9z=54$ intersects each coordinate axis.

For this I need to find points $(a,b,c)$ for the $x$-axis, $y$-axis, and $z$-axis. I know how to solve this when given two planes. First I'd set $x=0$ and have $y+9z=54$, but with only one plane I ...
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24 views

Relationship between $ A \mathbf{1} $ and$ A \mathbf{1}^{\perp} $

I have a square matrix $A \in \mathbb{R}^{n \times n}$ and I know that: $$ A \mathbf{1} = e_1$$ where $\mathbf{1}$ is the all ones vector and $e_1$ is the first element of the canonical basis. My ...
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0answers
8 views

Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
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2answers
35 views

Dimensions of $Pol(\mathbb{Z}_3)$ (polynomial vector space)

How many dimensions does $Pol (\mathbb{Z}_3)$ have, where $Pol (\mathbb{Z}_3)$ is a vector space of polynomial functions with one variable ($f: x \mapsto \sum^n_{k=0} \lambda_kx^k$). My "guess" would ...
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46 views

Dimension of the vector space defined by matrices product

Let $A$ be an $p×q$ matrix of rank a and $B$ a $r×s$ matrix of rank $b$. Please find the dimension of the vector space $M$. My attempt:I first product some elementary matrices on both sides of $ACB$....
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28 views

The norm of a multilinear transformation between finite-dimensional vector spaces is always finite

I am studying n-linear maps from Zorich, Mathematical analysis II, p. 49-53, where the author writes that "it is not difficult to prove that for mappings of finite-dimensional spaces the norm of a ...
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23 views

Identification of vector spaces

I am given an inclusion of vector spaces $L \subseteq V$ and I know that $V^{\vee \vee} = W^\vee$ for some vector space $W$. There are no assumptions of finite dimensionality. Through the inclusion $$ ...
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15 views

The sufficent condition about the direct sum and intersection of subspaces

Let $A,B,C,D$ be subspaces of a vector space over a finite field such that $\dim(A\cap B)=0$, $\dim(C\cap D)=0$, then what is the sufficent conditions that 1) $(A\oplus B)\cap(C\oplus D)=(A\cap C)\...
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21 views

Dual basis vectors of a normed space are bounded.

Let $V$ be a finite dimensional normed vector space over a field $\mathbb{F}$ with basis $\{e_1,...,e_n\}$, we know that for all $x\in V$ there are unique scalars $a_1(x),..,a_n(x)$ s.t $x=\sum_{i=1}^...

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