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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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How i x j= k (vector) BUT ixj = (i)(j) (sin90) = (1)(1)(1) = 1 (Scalar) [on hold]

How i x j = k (vector) , also in josiah willard Gibbs book who first given the idea of cross product did not explain the mathematical way of cross product. Also from quaternions i found no real ...
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32 views

Eigenvalue of a matrix $A$

WTS: A scalar $\lambda$ is an eigenvalue of a matrix $A$ $\iff$ $\det(\lambda I-A)=0$ My proof: Assume $\lambda$ is an eigenvalue of A. So $Av=\lambda v$ for a non-zero vector, v.This is equivalent ...
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2answers
10 views

Show that $A$ is a difference between two orthogonal projections.

Let $V$ be a finitedimensional complex vector space. Linear operator $A \in L(V) $ is hermitian and unitary. Show that $A$ is a difference between two orthogonal projections. The questions seems ...
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1answer
15 views

Basis formed by three non-coplanar vectors.

Suppose $\bf p, q, r$ are three non-coplanar vectors in ${\mathbb{R^3}}$. There is a vector $\bf x$ having projections along them are $a, b$ and $c$ respectively. Then can we write $$ \bf x = a\bf p + ...
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2answers
332 views

Given that A, B and C do not lie on the same line…find the area of triangle ABC.

\begin{array} { c } { \text { Given that } A , B \text { and } C \text { do not lie on the same line.If } \vec { O A } + \vec { O B } + \vec { O C } = 0 , | \vec { O A } | = \sqrt { 6 } \text { , } } \...
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2answers
46 views

Is an element of a vector a scalar or 1x1 vector or 1x1 matrix?

In econometrics, by Hayashi, they defined the error vector of n observations in a $ (n \times K)$ regression funcntion as: $\epsilon = \begin{bmatrix}\epsilon_{1} \\\epsilon_{2} \\\vdots \\\epsilon_{...
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2answers
24 views

The direction of a polynomial as an element of a vector space. [on hold]

By the definition of vector we mean a quantity which has a direction and magnitude but if we consider vector space of polynomials over some field then how direction of polynomial be defined
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Biconformal space and curvature

I've found very few contributions about the so called Biconformal Space, "a curved phase space". I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
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1answer
19 views

How to intepret the vector space of nxn Matrices over $\mathbb{R}$(real numbers) as an $ \mathbb{R}^{n^{2}} $vector space.

Let $ V=\mathbb{R}^{nxn} $ be the vector space of nxn-Matrices, in an exercise I need to interpret V as $ \mathbb{R}^{n^{2}} $. Can somebody explain me, how this is possible? PS: I'm not used to ...
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1answer
18 views

How to extract the screw axis vector and the angle from the exponential coordinates?

Given the 6-dimensional vector of the exponential coordinates of the homogeneous transformation: $S\theta$, where $S$ is the screw axis consisting of the pair $(\omega, v)$ and both of them are $3$ ...
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Let $V$ be a $n$-dimensional real vector space and let linear operator $ T \in L(V) $ satisfy the equation$ (T^2+I) *(T^2+4I)=0$.

Let $V$ be a $n$-dimensional real vector space and let linear operator $ T \in L(V) $ satisfy the equation $$ (T^2+I) *(T^2+4I)=0$$. Find the eigenvalues for $T$ and prove that $n$ is even. I'm a bit ...
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1answer
21 views

Prove that $\ker P(A^\ast) $ is an invariant subspace of $A$.

Let $A$ be a normal linear operator on a finite dimensional unitary space and $P(x)$ a polynomial. Prove that $\ker P(A^{*})$ is an invariant subspace of $A$ (where $A^{*}$ is its adjoint operator). I ...
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2answers
12 views

Finding normal operator matrix from characteristic polynomial

Let $ A \in L ( \mathbb{C}^4) $ be a normal operator with characteristic polynomial $ k_{A} = (\lambda - 1)^2 * (\lambda - 2)^2$. Is then the matrix for the operator just a diagonal matrix with ...
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1answer
20 views

Distance metric on vector space associated with edges of an undirected graph

Let $G = (V, E)$ be some graph representing for example, a physical road network. Intuitively, I can imagine that I can associate a distance between any two edges $e_1$ and $e_2$, $d(e_1, e_2)$, which ...
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13 views

Let $A$ be a normal linear operator on a complex finite dimensional unitary space which satisfies the equation:$ \tan A + \cot A = 2I$

Let $A$ be a normal linear operator on a complex finite dimensional unitary space which satisfies the equation: $$ \tan A + \cot A = 2I$$ Prove that $A$ has to be hermitian. I tried by looking at the ...
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1answer
6 views

Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent?

Let B be a nilpotent linear operator with index p. Is operator $ N^2 + \alpha*N $ nilpotent? ( $ \alpha \in \mathbb{R}$) I think that when $ \alpha = 0$ then the index is $p/2$ for even p, but when p ...
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3answers
36 views

Tangent vector field to a smooth curve over a smooth manifold

I am teaching myself some elementary differential geometry and am stuck on the concept of the tangent vector field of a smooth curve. I have searched the web for an hour or so but cannot find anything ...
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1answer
38 views

What does this equation imply?

I am reading a paper on seam carving to implement, and got stuck in understanding the convention used in the equation below. 1) I understand, $\{ ( x ( i ) , i ) \} _ { i = 1 } ^ { n } , \text { s.t....
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1answer
30 views

Complex Vector Spaces and Real Vector Spaces

Can any system that is represented using a complex vector space also be represented using a real vector space of double the dimensions?
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1answer
20 views

An other way of proving that : $(^tA A)_{i,j} = \langle a_i, a_j \rangle$

Thanks' to a recent question I know the following : If $A \in M_{n,k}(\mathbb{R})$ with columns $(a_1, ..., a_k)$ then $(^tA A)_{i,j} = \langle a_i, a_j \rangle$. It's very easy to prove since ...
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2answers
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Why is $U ^tU $ an orthogonal projection on $\operatorname{Im}(U)$?

Let $U \in M_{n,k}(\mathbb{R})$ such that : $^t UU = I_k$. Then I would like to understand geometrically why $U ^t U$ is the orthogonal projection on $\operatorname{Im}(U)$ ? When $n = k$ we are ...
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1answer
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Linear Algebra - Finite Dimensional Vector Spaces and their Rank [on hold]

Let U, V, and W be finite-dimensional vector spaces, and let $T : U → V$ and $S : V → W$ be linear transformations. Prove true or false: $1.) rank(S ◦ T ) ≤ rank(S)$ $2.) rank(S◦T)≤rank(T)$
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Unbiased real vector with respect to arbitrary orthonormal basis for a finite Hilbert space

Here I use Dirac notation to denote vectors. I would like to show that for an arbitrary orthonormal basis $\{ |\psi_k\rangle \}_{k=1}^n \subset \mathbb C^n$, $$\langle \psi_i | \psi_j \rangle = \begin{...
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0answers
24 views

On the characteristic polynomial

If $P_A(X) = \prod_{i = 0}^n (X-\lambda_i)^{n_i}$, where the $\lambda_i$ are distinct, then we know the following: $$\dim(\operatorname{Ker}(A-\lambda_iI)^{n_i})= n_i.$$ With this property in mind ...
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1answer
21 views

Span and Two Way Containment

I am proving : If $v_1, . . . , v_m$ and $w_1, . . . , w_n$ are vectors in V , then $Span(v_1, . . . , v_m)$ + $Span(w_1, . . . , w_n)$ = $Span(v_1, . . . , v_m, w_1, . . . , w_n).$ I am trying to ...
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0answers
19 views

Understanding of proof skeleton $\mathrm{vect}_k$

I am referring to the proof in http://mathserver.neu.edu/~jose/TannakianCats.pdf, Prop. 2.12. So far, I thought that one could not find an equivalence between the category $\mathrm{vect_k}$ and the ...
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1answer
43 views

Does the set of commuting endomorphisms always admit a cyclic endomorphism?

Let be $C(u)$ the set of commuting endomorphisms with $u$ over a complex-valued vector space of finite dimension $n \in \mathbb{N}^{*}$. I call $v$ a cyclic endomorphism iff all its eigenspaces are ...
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0answers
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Reproducing Kernel Hilbert Spaces

I don't undestand why: The set of continuos functions from a metric space X→R, C(X), forms a vector space over R using the usual definitions of addition and scalar multiplication.
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order on positive matrix

I know that the set of positive matrix (ie. The matrix $A \in M_n(\mathbb{R})$ such that $^tA = A$ and the eigenvalues of $A$ are $\geq 0$) is a convex cone. Thank do this we can defined a order ...
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0answers
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How to calculate angle between vectors between different bases?

I have one vector $ \vec{[v]_e} = (2,1,1)^T $ in natural base and transformation $ T(x,y,z) = (x-y,y-z,z-x) $ in natural base I also have two bases $$ B = \begin{pmatrix} 1 & 1 & 0 \\...
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1answer
41 views

Rudin's functional analysis, section 3.22 (Extreme points)

I struggle to understand the definition of "Extreme set" and "Extreme point". Let $K$ a subset of a vector space $X$. A non-empty set $S \subset K$ is called an extreme set of $K$ if no point of $S$...
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1answer
41 views

Condition for the projection on $S_n(\mathbb{R})$ to be orthogonal

For a matrix $B \in M_n(\mathbb{R})$ we denote $B_s = \frac{B+^tB}{2}$. Let $A \in M_n(\mathbb{R})$ such that all the eigenvalues of $A_s$ are in $[-1;1]$. Prove that there exist an orthogonal ...
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Cross product in index notation [closed]

Let $(\tilde{e}_{i=1}^3)$ be the standard basis of the $\mathbb{R}^3$.Let $(\tilde{g}_{i=1}^{3})$ arbitrary basis of $\mathbb{R}^3$. Show that $g_{i}\times g_{j}=\sqrt{\det g}\,\epsilon_{ijk}g^{k}$. ...
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1answer
55 views

How to show the solution of $\dot{x} = Ax $ is an invariant subspace?

Consider the linear dynamical system $\dot{x} = Ax $ in $V$ a finite dimensional vector space. The definition of an invariant subspace $U$ is as follows: For all $x_0 \in U$, (initial condition), ...
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1answer
21 views

Given a set and a basis of $\Bbb R^3$, find coordinates of vectors in the basis $B$ of the orthogonal complement of the set

Let $\Bbb S=\{\vec x\in\Bbb R^3\mid x_1-x_2=0\}$ and $B=\{(1,2,1),v,(2,1,1)\}$ be a basis of $\Bbb R^3$. It is known that $[(-3,9,1)]_B=\Bigl(\begin{smallmatrix}2\\3\\-1\end{smallmatrix}\Bigr)$. Find ...
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1answer
27 views

Subspace Criterion

Given V = {f :R→R | f(x)=a+bx+cx^2 where a, b, c ∈ R} and W = {f :R→R | f(x)=α+βx^2 where α, β ∈ R}. How do I prove that the subset W ⊆ V is a subset of vector subspace V? I know that W needs to be ...
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2answers
119 views

$AB-BA$ invertible and $A^2+B^2 = AB$ then $3$ divides $n$

Let $A, B \in M_n(\mathbb{R})$ such that : $AB-BA$ invertible and $A^2+B^2 = AB$ then prove that : $3 \mid n$. I tried to manipulate or find some factorisation in order to be able to use efficiently ...
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If I have $ {( \tan(A) - I)} ^{2} =0, $ for a linear operator on a complex finite dimensional space can I deduce from that $ \tan(A) = I $?

If I have $$ {( \tan(A) - I)} ^{2} =0, $$ for a linear operator on a complex finite dimensional space can I deduce from that $$ \tan(A) = I? $$ Moreover, is it then $$ A = \arctan(I)? $$
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Convergent sequence such that its components doesn't converge to the components of the limit

I would like to find a vector space $E$ a norm on $E$ a sequence $(u_n)$ which converges for this norm but such that its components doesn't converge to the component of the limit. First $E$ should ...
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1answer
25 views

How do I find a function that acts as the zero vector in a vector space

Question I understand that based on the axioms of vector spaces there needs to be a unique member the zero vector in V such that for all v element of V, v+0=v, but how do I find the appropriate ...
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2answers
42 views

Is every ideal in $K[x, y]$ of finite codimension necessarily prime?

I'm trying to answer the following question: Suppose that $R$ is an integral domain containing a field $K$. Then we may view $R$ as a $K$-vector space. Show that if R is finite dimensional as a K-...
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2answers
29 views

Confirming Axioms of Vector Spaces that rely on modular arithmetic

V is a vector space where $$V = \{\mathrm{rotations}\} = \{\theta : θ ~ \text{is a real number and} ~ 0 ≤ θ < 2π\}$$ Addition is defined by $$θ_1 + θ_2 := (θ_1 + θ_2) ~ \mathrm{mod} ~ 2π$$ ...
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2answers
31 views

how prove that a linear transformation is diagonalizable, given an eigenvalue and the dimension of its kernel

A question from an exam : (First year mechanical engineering, first course in linear algebra): Let $V$ be the vector space of $2\times2$ matrices, and let $U$ be the subspace of $V$ containing $2\...
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1answer
14 views

Linear Algebra Span Question with Specific Constraints

I received a question about a girl who can ride a hover board and a magic carpet whose restricted to move in the "movement vector" [3 1] and [1 2] respectively. The "movement" vectors represent how ...
3
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1answer
42 views

Find :$W_1 \cap (W_2 +W_3)$

let $W_1, W_2$ and $W_3$ be subspaces of $V$ Let $V=\mathbb R^2$ and $$W_1=\{(x,y)\in V:x=y\}$$ $$W_2=\{(x,y)\in V:x=0\}$$ $$W_3=\{(x,y)\in V: y=0\}$$ Then $W_1 \cap (W_2 +W_3)$ is equal ______? ...
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1answer
34 views

Vector space when the operations are non-standard

I am taking a linear algebra course and one of the things they have shown us is how to determine that V is a vector space according to eight axioms. When the operations for o-plus and o-dot are ...
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0answers
12 views

Compilation of known facts about projectors

I would like to make a compilation of useful properties about projectors (either orthogonal one and non-orthogonal one). Let $P \in M_{n,p}(\mathbb{R})$, $P^2 = P$. Here are the properties I know :...
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1answer
29 views

Do all linear transformations map the zero vector to the zero vector? [duplicate]

For example, a linear transformation T that maps between two vector spaces V and W. Does T map the zero vector of V to the zero vector of W? Is that a rule for linear transformations? Or no? I've ...
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1answer
25 views

open ball such that : $t \mapsto \operatorname{rank}( U+ tA)$ is constant where $^tU U = I_p$

Let $U, A \in M_{n,p}(\mathbb{R})$ such that : $^t U U = I_p$. Prove that the function : $t \mapsto \operatorname{rank}(U + tA)$ is constant on a neighborhood of $0$. I have solution when $n = p$ ...
0
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1answer
22 views

Prove a miscellaneous result regarding Vector Subspaces

If $U$ is a subspace of a vector space $V$, and if $u$ and $v$ are elements of $V$, but one or both not in $U$, can $u+v$ be in U? Can $cu$ be in $U$ for some nonzero scalar $c$ if $u$ is not in $U$? ...