For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag: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 ...

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What is the formal mathematical representation of a “force”?

In mechanics, it is usual to represent a force by a 3-vector. When it is necessary to consider the turning effect of a force, the 3-vector is commonly "attached" to a point on its line of action. In ...
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29 views

The number of real values for which set is Not a basis of $R^{2} $

Given set is $(a, a^{2}) , (a^{2} , a ) $ . As i see that for a =1 ,0 and -1 set is not a basis .But how do i check it mathematically , if there are any other values ? Thanks
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Change of basis matrix for polynomials?

I've understood what a change of basis matrix is, and how it's structured. So a change of basis matrix from $B$ to $C$ is the matrix $M$ such that: $${\begin{bmatrix} &\\ \\ \\\end{bmatrix}}_B ...
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35 views

Prove that $\det(\text{Id}+T)\ge 1+\det(T)$

Let a self-adjoint operator $T:V\to V$ above $\mathbb{C}$, such that $\langle Tv,v \rangle \ge 0$ (so it's essentially a real number). We have learned before that for this kind of $T$, all it's ...
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1answer
21 views

Kernels of integral transform and linear transformation

Is there any relation between the $kernel$ of an $integral \ transform$ and the $kernel$ of a $linear \ transformation$?
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90 views

How to prove a type of functions is a subspace of the vector space of all functions.

I've been working on algebra and want to know how to determine if a certain type of function is a subspace of the vector space $\mathbb{R} \to \mathbb{R}$. So far I've been using the two properties ...
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2answers
159 views

hypothetical 4 dimensional vector space

Before I start asking the question, I want to apologize for my illiteracy in latex maths and the abstraction of my question. My question is, is it possible for me to define a hypothetical 4 ...
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20 views

what exactly is a center invariant subspace?

I'm studying robust control, and I have a matrix with two invariant subspaces. One is stable, which I assume is spanned from the eigenvectors with real part less than zero, and the other is a center ...
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32 views

Automorphism group of vector space

I was trying to understand definition of representation and trivial representation thus came across the case where $ V= K $ here $V$ is a vector space over a field $K$ and thus $Aut_K (V) \cong ...
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1answer
32 views

Linear Algebra Subspaces questions [closed]

Let V be a vector space over a field F and M,N≤V. Let M be the plane x+y+z=0 and N be the line x=y=z in R^3 . Show that R^3=M+N . (M+N is direct sum)
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To find dimension of subspace

Let W be subspace spanned by$ (2i,0,1,2i) , (0,2i-2,i-3,0), (-i,1,0,i), (1,1,1,1) $ I have tried to reduce it to RREF , but its such a pain . Is thereany shorter way i could do this ...
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1answer
25 views

Find equation for an isomorphism such that

In $\mathbb{R}^4$ plane V is given, $V=span(\alpha_1,\alpha_2)$ where $\alpha_1=[1,3,4,1]$, $\alpha_2=[1,2,2,3] $ a) Find the formula for isomorphism $\varphi:\mathbb{R}^4\rightarrow\mathbb{R}^4$, ...
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1answer
23 views

Change of basis with a nonlinear operator

Given a vector space $V$ and its two basis: $\mathcal{B}$ given by vectors $\{e_i\}$ and $\mathcal{B}'$ given by vectors $\{e'_i\}$, why are the two basis necessarily connected by a linear ...
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22 views

About hermiticity implying non-nilpotency for infinite dimensional vector space(Corrected)

Suppose $A$ is a non-trivial linear operator acting upon infinity dimensional vector space. Say given $A^2$=0 and provide that $A$ is hermitian. Is this sufficient to conclude $A$ is non nilpotent? I ...
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1answer
37 views

To determine basis for $V \cap W$

Let $V$ be vector subspace of$ R^{4} $ spanned by vectors $( 1,1,1,-1)$ and $(1,-1,0,1)$. Let $W$ be another vector subspace of $R^{4}$ spanned by $(1,1,-1,1)$ and $(1,3,4,-5)$. Determine basis of ...
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1answer
40 views

Prove that a set of matrices is a linear space

Prove that the set of matrices $$v:=\left\{ \begin{pmatrix} 2x-y+z & x-2y-2z \\ x+y-z & 3x+y+2z \end{pmatrix} \middle|\, x,y,z \in R\right\}$$ Is a linear space above $R$ and find it's base. ...
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35 views

Prove two simple projection statements

Let $U$ be a subspace of $\mathbb{C}^n$ and suppose $v \in \mathbb C^n$. Let p be the projection of the vector $v$ onto the subspace $U$. That is, p is defined as $P_U(v)$. Prove : (1) $\langle ...
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2answers
68 views

Why study real vector spaces of dimension $n$ other than $\Bbb{R}^{n}$?

Every vector space of dimension $n$ is isomorphic to $\mathbb{R}^{n}$. Why do we study other (finite-dimensional, real) vector spaces?
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34 views

Minkowski sum and difference do not cancel

I need to prove that $ ( A\oplus B )\ominus$ $B$ and $( A\ominus B)\oplus$ $B$ need not equal $A$ for all sets $A$, $B$,where $\oplus$ and $\ominus$ denote the Minkowski sum and difference. As ...
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14 views

How to determine a supplementary subspace?

Problem: Determine a supplementary subspace $U$ to the subspace $W = span\left\{(1,6,4)\right\}$ of $(\mathbb{R}, \mathbb{R}^3, +)$. Determine also for the vectors $v = (2,2,0)$ and $w = (0,4,4)$ ...
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1answer
32 views

Schur's Triangularization Lemma in Hefferon's Linear Algebra textbook

I'm reviewing some material and came to this: Fix a basis $B = \{\vec{\beta}_1, \ldots, \vec{\beta}_n\}$ for $V$ ($V$ is a vector space) and observe that the spans $$ [\emptyset] = \{\vec{0}\} ...
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1answer
29 views

How to express the Pythons' NumPy linspace or arange arrays mathematically?

How one can express digital one dimensional array, such as x = np.linspace(0, 10, 1000) or x = np.arange(-1, 1, 0.01) (examples ...
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1answer
34 views

Does there exist an infinite dimensional vector space over an infinite ordered field which cannot have any inner-product imposed on it?

The title says it all. I'm wondering if there is any infinite dimensional vector space over some infinite ordered field such that we cannot impose any inner product on it at all. I understand that ...
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1answer
55 views

Can i Find the Matrix from Eigenvalues and Eigenvectors?

If i given eigenvector: $$V_1=\begin{pmatrix} {1\over \sqrt{3}}\\{1\over \sqrt{3}}\\{1\over \sqrt{3}}\end{pmatrix} , V_2=\begin{pmatrix} {1\over \sqrt{6}}\\{-2\over \sqrt{6}}\\{1\over ...
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1answer
41 views

what is the geometical interpretation of $\vec a.\vec b$? [duplicate]

what is the geometical interpretation of $\vec a.\vec b$?(dot product) I know the projection of $\vec a $ on $\vec b$ is $\vec a.\hat b$. But what is a projection here?
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1answer
18 views

Qestion about Eigenvector, basis for the solution

I'm confused with some question currently I'm trying to solve. If you help that will be grateful. Given the matrix find eigenvalues and eigenvectors $$ A = \begin{bmatrix} 4 & -2 ...
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1answer
30 views

Solution space of a Differential Equation

Generally, initial conditions to an $n^{th}$ order ODE involve initial conditions only involving derivatives up to the degree $ n-1 \ (like \ y^{(n-1)}(0) \ = \ A).$ Even a basis of the space of ...
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0answers
40 views

What makes a norm-Gaussian inner product space “infinite-dimensional”?

Suppose we define an $\mathbb{R}^m$ inner product space in which the inner product of $\mathbf{x}$ and $\mathbf{y}$ is $\exp\left(-\|\mathbf{x} - \mathbf{y}\|\right)$. In PCA and machine learning, we ...
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1answer
71 views

Show that $V^*$, set of all Linear Transformations from $V$ to $R$, is a vector space

$V$ is a vector space, and $V^*$ is the set of all LT's from $V$ to $\mathbb{R}$. a) Show that $V^*$ is a vector space. b) Suppose $\{v_1,\dots,v_n\}$ is a basis for $V$. For $i = 1,\dots ,n$ define ...
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41 views

Prove that there exist $W$ such that $V=V_1\oplus W=V_2\oplus W$

Let $V$ be a finite-dimensional vector space. If $V_1$ and $V_2$ are distinct linear subspaces of $V$ such that $\dim V_1=\dim V_2$, show that there exists a linear subspace $W$ of $V$ such that ...
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1answer
22 views

Extreme points of complex sphere of dimension n in 1-norm.

I came up with the following question while learning about different norms in $\mathbb{C}^n$. For $z=(z_1, \ldots, z_n)^T \in \mathbb{C}^n$ we consider the 1-norm: $\|z\|_1= \sum_{k=1}^n|z_k|$. Let ...
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1answer
186 views

Confusing notation $D(p)(x)$ in a vector space of polynomials

If we have a vector space that consists of all polynomials of degree less than or equal to 4, and we consider the following function: $$D(p)(x) = 2.5\cdot p(x-1)$$ where $p$ is a function from the ...
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3answers
46 views

Are there any simple/explicit examples of a finite vector space?

By finite vector space, I mean a non-trivial vector space with a finite number of elements, not just a finite field. I'm hoping for a really simple example, even better if that set is explicitly ...
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1answer
31 views

Base of vector space from a finite set [closed]

Let $V$ be a vector space of finite dimension. $S=\{v_1,...,v_r\} \subset V$ and $Span(S)=V$. For each $v_i\in S$ there is a linear combination from $S\setminus \{v_i\}$. How can I show that for each ...
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3answers
67 views

To determine Nullity of $T$

Let $V$ be vector space of polynomials of degree $\leq n$ . And $ T : V \rightarrow \mathbb R ^{m}$ be defined as $T (P (x)) = (P (1) , P (2) ,..., P (m) )$ I have to determine nullity of $T$ . ...
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1answer
18 views

Finding a basis and the dimension of $W_1\cap W_2$

Suppose $W_1,W_2$ are subspaces of $\mathbb{R}^4$. $W_1$ is spanned by $(1,2,3,4), (2,1,1,2)$ and $W_2$ is spanned by $(1,0,1,0),(3,0,1,0)$. I have to find a basis for $W_1\cap W_2$. I have ...
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1answer
38 views

Dual basis in a finite separable extension

I am reading the book Algebras, Rings and Modules, volume 1, by M. Hazewinkel and at the page 193 there is a proof about why the integral closure of a ring in a separable finite extension L over $k$ ...
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1answer
31 views

Modules isomorphism

Studying vector spaces, we can findthe well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
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36 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
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1answer
36 views

A real vector space with a complex structure is naturally a complex vector space

I am struggling with this exercise from the book 'Tensors and Manifolds: With Application to Physics', by Robert H. Wasserman: Corresponding to each $a \in \mathbb{K}$ there is a linear operator ...
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1answer
29 views

If M is a closed subspace of X and $x ∈ X-M$ then $M + \mathbb Cx$ is closed.

Let X be a normed vector space. If M is a closed subspace of X and $x ∈ X-M$ then $M + \mathbb Cx$ is closed. where $M + \mathbb Cx=\{y+\lambda x:y\in M, \lambda\in \mathbb C \}$ the question ...
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69 views

Vector spaces and dimension: unordered pairs

Let $K= \mathbb{Z}_p$, where $p$ is a prime number, and let V be a vector space over the field K such that $\dim{V} = 3$. I have no idea where to start with this, I'm not even really sure what I'm ...
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5answers
530 views

Why do bases of infinite dimensional spaces need to be orthonormal?

I asked this question following a discussion in my Mathematical Methods course and didn't get a satisfactory answer. If we have an infinite dimensional Hilbert space, why do we need an orthonormal ...
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0answers
38 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.9, Problem 12

If $f_1, \ldots, f_p$ are linear functionals on an $n$-dimensional vector space $X$, where $p<n$, then how to show that there is a vector $x \ne 0$ in $X$ such that $f_1(x) = 0, \ldots, f_p(x)=0$? ...
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2answers
27 views

Smallest angle to turn

I have a an object that starts an arbitrary heading in degrees. This object will rotate about an angle to reach a target heading. To reach this target heading, you can rotate about two different ...
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1answer
48 views

Smallest angle between two vectors?

I have a robot and I am going to turn it clockwise (negative degrees) or counterclockwise (positive degrees). If I turn the robot -270 degrees, that is the same as turning +90 degrees. Is there a way ...
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1answer
35 views

Standard basis of a Matrix with identical entries.

How would you represent a $\mathbb{R}$-Matrixspace which looks like this $$\begin{bmatrix} a & b \\ b & a \end{bmatrix}$$ with standard basis I can't think of anything else but ...
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1answer
39 views

Uniform continuity of scalar multiplication in topological vector spaces

If $X$ and $Y$ are topological vector spaces over $\mathbb R$, then a map $f:X\to Y$ is called uniformly continuous if for each neighborhood $V\subseteq Y$ of $0\in Y$, there exists a neighborhood ...
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7 views

orbital behavior of objects in a space-time contraction field

I would like to find the orbital velocities of non-gravitating objects imbedded in a space-time contraction field. The field has the form as shown in figure 1. The surface space-time compression ...
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1answer
15 views

$A(2,1,3),B(3,2,4),C(1,8,9), D(4,3,12).$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.

$A(2,1,3)$ $B(3,2,4)$ $C(1,8,9)$ $ D(4,3,12)$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$. I am not sure how to calculate this. How do I calculate the ...