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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. In other words, these are the spaces where we can make sense of linear combinations.

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1answer
17 views

how to represent a vector

I have to define a state space equation. there is a vector with 5 members as stats such that all states(members of the vector) are real numbers. which notation is true for this case? x=[x1,x2,x3,x4,...
2
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2answers
24 views

Linearly dependent system of vectors with sum of coefficients $0$

Let $V$ be a vector space over a field $K$ so that $dim(V) = n$. Let $e_1, e_2,\dots,e_{n+2}$ be a vector system. Prove that there exists $k_1,k_2,\dots,k_{n+2} \in K$ which are not all $0$ so that $$ ...
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0answers
8 views

N-driven Vector Space

Find the no of BASES of Vector Space V (n-dimensional) over a finite field of Order ‘q’
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0answers
34 views

Difference between $x^{T}Ay$ and $y^{T}Ax$ when $A$ is not symmetric?

$x^{T}Ay=y^{T}Ax$ if $A$ is symmetric. what is the difference between the two when $A$ is not symmetric? Is the difference negligible under some condition?
0
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1answer
31 views

Using partial derivatives to find normal vector

So, I can not find out what I'm doing wrong with this question, even if my life depended on it. I know instruction for doing it, but I can't seem to figure out what I'm doing wrong. Because I refuse ...
0
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2answers
34 views

Corresponding space to a irreducible part of a given representation of $S_n$

My setting is the following: I have an action of $S_n$ on some $\mathbb{C}$-vector space, $V$, by permuting a special basis: $$<a^1_{1},\dots,a^1_{n!},a^2_{1},\dots,a^2_{n!},\dots,a^k_{n!},b_1,\...
0
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1answer
27 views

A linear transform mapping one billinear map to another may not exist (counter-example)

$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as $$ N_1(\phi) = \{ v \...
0
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0answers
10 views

skew-symmetric non-degenerate bilinear space has even dimension [duplicate]

How to prove skew-symmetric non-degenerate bilinear space has even dimension with skew-symmetric defined as $(x,y) = - (y,x), \forall x,y \in V$, where $V$ is the vector space
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1answer
33 views

Vector Space with Hamel Basis is not Separable when all basis elements are 2 apart

Consider a Hamel basis, $\{e_{\lambda}\}_{\lambda \in \Lambda}$, for an infinite dimensional linear vector space. I'm reading something that makes the following claim in passing: Note that if for ...
1
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2answers
31 views

What does it mean to treat $\mathbb{C}$ as a real vector space?

My textbook says (on PDF page 27): "The set $\mathbb{C}$ of complex numbers can be canonically identified with the space $\mathbb{R}^2$ by treating each $z = x+iy$ as a column $(x,y)^T \in \mathbb{R}^...
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0answers
20 views

Is the norm of a vector given by dividing it by the square root of the inner product with itself?

I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product. But can this be ...
1
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1answer
25 views

For A,B subsets of a Normed Vector Space, A closed, B Compact, Show A - B Is Closed [duplicate]

Statement of the problem: Let $E$ be a Normed Vector Space over the real numbers. Let $A, B$ be subsets of $E$ such that: $A$ and $B$ are non-empty, $A \cap B = \emptyset $. Assume $A$ is closed and ...
0
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0answers
11 views

XYZ 2 Length Rotation Angle (YBC) [on hold]

XYZ2YBC(X,Y,Z,R) where R is the bend radius. convert the X,Y,Z values to Y,B,C values, taking in to account the bend radius R
0
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1answer
39 views

The eigenspace of $\lambda$ is simply the null space of the matrix $A-\lambda I$. Is that correct?

Would it be correct to say that the eigenspace corresponding to an eigenvalue $\lambda$ for a matrix $A$ is simply the null space of the matrix $A-\lambda I$? My justification is that the eigenspace ...
0
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0answers
29 views

Find a basis for the row space, column space, kernel, and image of the following matrix verification

For the following matrix: $$ \begin{bmatrix} 1 & 2 & 1 & 3 \\ 2 & 5 & 5 & 6 \\ 3 & 7 & 6 & 11 \\ 1 & 5 & 10 & 8 \\ \end{bmatrix}...
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2answers
38 views

Find distance between a line and the origin [on hold]

The given line is defined as following: $$\text{plane 1: } x+y+z = 6$$ $$\text{plane 2: } 2x - y - 5z = -5$$ What is the distance between the line and the origin?
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0answers
44 views

Can real vector space be a model of first order logic? [on hold]

As far as I understand the linear vector algebra is first order theory, some instance of first order logic. But what about generality? Can ve construct real (or trascendental, hypercomplex or polyadic)...
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0answers
23 views

Is the set of 2-vectors having a zero in at least one coordinate a subspace of $\mathbf{R}^2$

Here is what I did: 1) ${V}*{0} = {[0,a]}*{0}={[0,0]}$ 2) ${[0,a]}+{[0,b]} = {[0+0,(a+b)]}$ 3) ${c[0,a]} = {[c*0,c*a]} = {[0,(c*a)]}$ So then this is subspace of $\mathbf{R}^2$ . Is this correct? ...
1
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1answer
53 views

Dimension of kernel subspace of trace transformation

From S.L Linear Algebra: Let $V$ be the vector space of real $n \times n$ symmetric matrices. What is $\textrm{dim} \, V$? What is the dimension of the subspace $W$ consisting of those matrices ...
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0answers
41 views

Functions as Vectors

Whenever I refer a book or video on how to represent a function as a vector, the source automatically assumes the function to be a polynomial $$a_0 + a_1 \alpha +a_2 \alpha^2 + ... + a_n \alpha^n $$ ...
0
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2answers
60 views

Denoting vector spaces

Let $X_i$ from $1 \leq i \leq n$ be vector spaces. Now I want to denote a space $X$, such that each $x \in X$ is given by $x = (x_1,...,x_n)$, and $x_i \in X_i$. Similarly, for each possible vectors, $...
0
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1answer
33 views

Showing 2 matrices span the same subspace?

Lets say we have a $m \times n$ matrix $A$ and a $m \times n$ matrix $B$. $A$ and $B$ span the same subspace if and only if there is a $n \times n$ matrix $C$ such that $B$ = $AC$. Show that $A$ and ...
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0answers
24 views

Meaning of “Fq subspace of Fqm” [on hold]

Can someone please explain what this "Fq subspace of Fq^m" is. See the image for context. enter image description here Sorry. I'll elaborate. Here 'q' is a 'p^r' where p is a prime and r is an ...
0
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1answer
28 views

Understanding the symbol <A,B> in affine spaces

I'm trying to solve this exercise: A subset F of an affine space is an affine subspace if and only if for all points A and B of F, the inclusion <A, B> ⊂ ...
0
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0answers
44 views

Showing that the solutions of the differential equation $u''' - 3u'' + 4u = 0$ form a vector space.

My textbook has the following problem: Show that the solutions of the differential equation $u''' - 3u'' + 4u = 0$ form a vector space. Find a basis of it. This is a third-order linear homogeneous ...
0
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3answers
25 views

Show that S is a subspace of ${R}^{2\times2}$

Let v= (1,2)$^T$ be a given vector, and let $S$ = {$A$ ∈ ${ \mathbb{R} }^{2\times2}$ | a$_1$$\bot$v}. (I.e., $S$ is the set of all 2x2 real matrices with column 1 orthogonal to the given vector v.) ...
1
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0answers
18 views

On a cartesian description of the null space of a matrix

Given a matrix A, I was taught that to find a cartesian description of the null space of the matrix, we would find the basis of the row space (which is the transpose of the rows with leading ones in ...
1
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4answers
44 views

Define dimension without referring to bases

The dimension of a vector space is the common cardinality of all bases. I would like to define it in a way that does not refer to bases. I only care about finite dimensional vector spaces. First, I ...
1
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1answer
34 views

How to get a basis of $U \cap V$ where U and V are the column space of $A$ and $B$.?

Problem: Let $A = \begin{bmatrix}5& 2& -1\\3& 1& 0\\ -1& 0& -1 \end{bmatrix}$, $B = \begin{bmatrix} 4& -3\\ -2& 3\\ 1& -2\end{bmatrix}$, $U = C(A)$ and $V = ...
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2answers
39 views

Equivalence between these tensor product definitions

Let $V$ and $W$ be vector spaces. Then the tensor product $V \otimes W$ of $V$ and $W$ is the vector space $V \otimes W$ together with a bilinear map $\phi: V \times W \rightarrow V \otimes W$ such ...
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0answers
13 views

Show that $L_{p,\infty}(0,\infty)/L_{p,\infty}^0(0,\infty)$ is a reflexive space [on hold]

Let $L_{p,\infty}(0,\infty)$, $1<p<\infty$, be the weak $L_p$-space on $(0,\infty)$ and $L_{p,\infty}^0(0,\infty)$ be its separable part (the closure of all bounded functions having finite ...
6
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1answer
50 views

Is the differential at a regular point, a vector space isomorphism of tangent spaces, also a diffeomorphism of tangent spaces as manifolds?

Note: My question is not "If $f$ is a diffeomorphism, then is the differential $D_qf$ an isomorphism?" My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I didn't study much of ...
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0answers
14 views

Reference request for bigraded vector spaces

I'm currently reading up on Spectral Sequences in Algebraic Topology, and often times authors refer to graded vector spaces and bigraded vector spaces freely without defining them. I've found ...
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0answers
20 views

Define a function that operates on n-dimensional real space with different length

I'm racking my brain over a definition of a function for some problem. In general, I have the following problem that I want to describe mathematically. Lets say I have $m$ bins that consists of $k$ ...
1
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1answer
29 views

Can a subset not belonging to $\mathbb R^n$ still be contained in a subspace different than $\mathbb R^n$?

So basically I have the following subset $$W = \{(x,x,xy,y,y)\} : x,y \in \mathbb R\}$$ that, after checking if it is closed to addition and multiplication I realize it is not a subspace of $\mathbb R^...
4
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2answers
192 views

Are two submodules (where one is contained in the other) isomorphic if their quotientmodules are isomorphic?

Let $M$ be an $R$ module and $N_1 \subset N_2$ be submodules of $M$ such that $M / N_1 \cong M / N_2$. Can I know conclude $N_1 \cong N_2$ or even $N_1 = N_2$? I know that a proper submodule can be ...
0
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0answers
16 views

Differentiable at every point in normed vector space

How can I prove that if $V$ and $W$ is normed vector spaces and $T \in LC(V; W)$, $T$ is differentiable at every point? And how to find its derivative? I have no idea how to start and what theorem to ...
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3answers
42 views

Let $w_1 = (0,1,1)$. Expand {$w_1$} to a basis of $R^3$.

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section3.5 Exercise 7, I was puzzled at some of it. Here is the problem description: Let $w_1 = (...
0
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0answers
31 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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0answers
6 views

Unit vector Identicon

Is there a way to visualize multiple n-dimensional ($n\approx300$) unit vectors with the property that large changes to the vector result in large visual changes, but small changes result in little to ...
0
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2answers
24 views

A puzzle about replacing $v_1$, $v_2$, $v_3$ while retaining the linear independence of the resulting set.

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section3.5 Exercise 5, I was puzzled at some of it. Here is the problem description: Exercise 5. ...
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0answers
23 views

How to find the location of a point in 3D space from projected 2D angle

I have points $A,B,C$ in 3D space and I know the position of $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. I want to find the location for $C$ given that $AB$ and $BC$ is perpendicular in 3D space but ...
1
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1answer
38 views

Axler's book $ F^{\infty} $

In Axler's book he defines lists and says that its' lenght has to be finite by definition each list has a finite length that is a nonnegative integer. Thus an object that looks like (x1; x2; ...
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0answers
18 views

$W = \{ \alpha \in \mathbf{R}^n | \alpha' A \alpha = 0\}$ doesn't have a subspace of more than $n-p+1$ dimensions.

If $A$ is a full rank symmetric matrix of real numbers $R$, prove that : $W = \{ \alpha \in \mathbf{R}^n | \alpha' A \alpha = 0\}$ doesn't have a subspace of more than $n-p+1$ dimensions($p$ is the ...
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0answers
20 views

The underlying group of a positive-dimensional vector space is not free abelian

This is homework: "prove that the underlying group $G$ of a positive-dimensional vector space is not free abelian". My solution is the following. If the field of definition has positive ...
0
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2answers
28 views

Given a space of 3x3 matrices and a subspace W, how to determine if there is a subspace such that $W \oplus U = V$

Let W be a set of matrices of the form $$ \begin{bmatrix} A & B & C \\ B & A & B \\ C & B & A \\ \end{bmatrix} $$ where A, B, C are real numbers and W is a ...
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0answers
12 views

Given a smooth map $\sigma$, and a linear isomorphism $K$, is there a smooth map $\tau$ s.t $D(\tau \circ \sigma^{-1}) = K$

Let $(M, \Sigma)$ be a smooth manifold, and $\sigma, \tau$ be two smooth charts defined on the neighbourhood of $x_0 \in M$. Then by definition $$\tau \circ \sigma^{-1}$$ is a diffeomorphism from an ...
0
votes
2answers
29 views

Line segments are always compact?

A line segment from $x_0\in X$ to $x_1\in X$ is a subset of the vector space $X$ given by $\{x_0+tx_1: \ t\in [0,1]\}\subseteq X$ I know that this result is true for finite-dimensional vector spaces ...
0
votes
1answer
63 views

What is $\deg(f)$ when $\int_M \omega = 0$, if that's possible?

In this question, Jrrow assumes $\int_N\omega_0\neq 0$ Why is $\deg(f)$ well-defined? What if $\int_N\omega_0\neq 0$? My book does not seem to address this explicitly. If $\int_N\omega_0 = 0$, then $...