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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 ...

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25 views

Prove that a bilinear form is nondegenerate

Let $n$ be a positive integer, $V=\mathbb{C}^{n\times n}$ and $$f(A,B)=n\, Tr(AB) - Tr(A)Tr(B)\quad .$$ Let $$W=\{ A\in V\, |\, Tr(A)=0 \} $$ and let $f_1=f|_{W}$. Prove that $f_1$ is nondegenerate....
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7 views

Killing form for a different representation

In this Math Overflow question, the OP was asking if the Killing form defined for a different representation (than the adjoint) was related to the normal Killing form (defined w.r.t. the adjoint rep.):...
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1answer
12 views

Image of a Polynomial Basis

Let $B$ $=$ {$x^2, x, 1$} and $S$ $=$ {$x^2+x, 2x-1, x+1$} be two basis of $P_2$. Let $T$ be a linear transformation from $P_2$ to $P_2$ such that the transition matrix from $B$ to $S$ is $\begin{...
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1answer
17 views

Can we say that vector of branches spans $(n-1)$ dimension space or incidence matrix rows (columns?) span $(n-1)$ dimension space?

If we have connected graph and $\mathbf\Phi=(\Phi_1,\Phi_2...\Phi_n)$ - nodes, and $\phi_k=\Phi_i-\Phi_j$ so $\mathbf{\phi}=(\phi_1,\phi_2...\phi_N)$ - branches (edges), can we say that vector $\...
2
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0answers
39 views

divergence in polar coordinates

For a vector field $X$, the divergence in coordinates is given by $\nabla\cdot X=\sum_n\frac{X^i}{\partial x^i}$. In polar coordinates, the metric is $\begin{bmatrix}1 & 0\\ 0 & r^2\end{...
3
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2answers
51 views

$\ell^2$ as colimit in $\mathbf{TopVect}_{\mathbb{R}}$

Let $\mathbf{TopVect}_{\mathbb{R}}$ be the category of topological vector spaces with continuous linear maps as morphisms. Is it ineed true that $\ell^2 \cong \varinjlim_{n}\oplus_{i=1}^n\mathbb{R}$?
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21 views

Homogeneous ODE Multiplicative-Additivity Distributivity Axiom

Problem Suppose q(x) and p(x) are continuous functions. Show that the set of all solutions to the following ODE gives rises to a vector space. $$y'' + py' + qy = 0$$ Let $y_1$ and $y_2$ be linearly ...
6
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2answers
456 views

Find locus of points by finding eigenvalues

Let $\boldsymbol{x}=\left(\begin{matrix}x\\ y\end{matrix}\right)$ be a vector in two-dimensional real space. By finding the eigenvalues and eigenvectors of $\boldsymbol{M}$, sketch the locus of points ...
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2answers
31 views

Determine all real numbers that can be eigenvalues of operator A.

Problem: Let $A$ be some linear operator such that: $$ (A^{2006}-I)^{2006}-I=0. $$ Determine all real numbers that can be eigenvalues of operator $A$. Question: How to solve this problem? My ...
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Linear algebra - vector space - norm [on hold]

Let $X$ be a complex vector space with an inner product $\left \langle \cdot,\cdot \right \rangle $ , defining a norm on $X$ by usual formula $$\|x\|=\sqrt{\langle x,x\rangle}\;,\;\;x\in X$$ Show: ...
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1answer
19 views

How does hyper-plane equation divide vector space into cells convex polyhedrons?

I came across some specific algorithm that divides high-dimensional vector space into non-overlapping cells of convex polyhedron. It does this by using tree based binary partitions (which might not ...
4
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1answer
65 views

Showing any linear operator $T : X \to Y$ is bounded, where $X$ is a finite dimensional normed vector space, and $Y$ any normed vector space.

Let $X$ be a finite dimensional normed vector space and $Y$ an arbitrary normed vector space. Show that any linear operator $T : X \to Y$ is bounded. I got the hint to first show that $\| x\|_0 := \| ...
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2answers
19 views

Finding the span of a subspace if I have its image as a matrix

This is at extract from a written solution to a problem: $$ U_1 = \operatorname{Im} \begin{bmatrix} 2\cos^2 (\theta/2) & 2\sin(\theta/2)\cos(\theta/2)\\ 2\sin(\theta/2)cos(\theta/2) & 2\...
4
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2answers
26 views

Subspace basis of $\mathbb{R}^n$ only with positive values

It seems obvious but I didn't find a proof yet: Let $U$ be an arbitrary subspace of $\mathbb{R}^n$. Set $m:=\dim{U}$. Can $U$ be written as $U=\mathrm{span}\{b_1,\dotsc,b_m\}$, $b_j=\begin{pmatrix}...
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4answers
29 views

Express vectors on a dodecagon

$P_1,P_2, ... P_{12}$ are consecutive vertices of a regular polygon with $12$ sides. If $\overrightarrow{P_1P_2}=\vec{x}$ and $\overrightarrow{P_1P_3}=\vec{y}$. Express the following vectors in terms ...
3
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1answer
28 views

$T$ and $U$ have a common eigenvector, where $T$ and $U$ are linear operators on odd dimensional vector space $V$ and $T^2 = U^2 = I$.

I am trying to prove that $T$ and $U$ have a common eigenvector, where $T$ and $U$ are linear operators on odd dimensional vector space $V$ and $T^2 = U^2 = I$. I have been stuck on this problem for ...
2
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2answers
48 views

How to proof that the $(M^\perp)^\perp = \operatorname{span}(M)$? [on hold]

I have some problems with the following problem. It seems to be so obvious that I don't just how to show that that is true. Could someone help me? Let $M \subset \mathbb{R}^n$ be a nonempty subset....
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0answers
32 views
+100

Generalize exterior algebra: vectors are nilcube instead of nilsquare

The exterior product on a ($d$-dimensional) vector space $V$ is defined to be associative and bilinear, and to make any vector square to $0$, and is otherwise unrestricted. Formally, the exterior ...
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2answers
48 views

Meaning behind $\mathbf{V} \subset \mathbb{R}^n$

When I have a vector space spanned by a set of vectors:$$\mathbf{V}=\mathrm{span}\{(a,b,c,d)^T,(d,e,f,g)^T,(2a,2b,2c,2d)^T\},$$How can I write $\mathbf{V}$ in terms of a subset to another vector space?...
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0answers
25 views

About Sobolev Space definition.

I would like to know if this two definitions for Sobolev Spaces are equivalent: $H_2^m([a,b]):=\{f: f',...,f^{(m-1)} \text{absolutely continuous}, \int _a^b (f^{(m)})^2dx<\infty \}$ and $H_2^m([...
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1answer
13 views

Trouble understanding the definition of a subspace

This is a very simple, probably silly, question, but I'm having a diffuclty in unerstanding the terminology used in definition of a subspace. From Wikipedia: If $V$ is a vector space over a field $...
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1answer
19 views

Angle between two intersecting lines that appear on a cut face

I am given the equations of 2 thin planes in a rock. Plane 1: $\mathbf r\cdot(0,3,-1) = -1$ Plane 2: $\mathbf r\cdot(1,0,1) = 1$ The entire set (the rock) is cut by the plane: $\mathbf r\cdot(-1,3,...
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3answers
66 views

In $\mathbb{R}^n$, is the dot product the only inner product?

Just what the title says. I'm reading, from various resources, that the inner product is a generalization of the dot product. However, the only example of inner product I can find, is still the dot ...
2
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2answers
20 views

General method to find the perpendicular distance between a plane and a point.

Here's the question I'm puzzling over: $\textbf{Find the perpendicular distance of the point } (p, q, r) \textbf{ from the plane } \\ax + by + cz = d.$ I tried bringing in the idea of a dot product ...
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0answers
57 views

Bizzare Hypothesis

Consider the equations: $ ax + by + cz = q $ - equation(1) $ dp + eq + fr = q $ - equation (2) where a, b , c , d, e ,f are scalars Given {x , y , z} is linearly independent And {p , q , r} is ...
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1answer
15 views

Two Change of Basis Matrices

Let $V$ $=$ $P_2$ and $dim(v)$ $=$ $3$. Let $B$ $=$ {$1$, $x$, $x^2$} and $S$ $=$ {$1+x$, $2-x$, $3+x^2$} be two basis of $V$. What is the matrix $P_B,_S$ and $P_S,_B$? I am not entirely sure how to ...
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1answer
18 views

Linear transformations - 2 opposite claims, solution attempt included

Suppose $V, W$ are vector spaces and $ T: V \to W $ is a linear transformation. $v_1, v_2, ... , v_k \in V$. Prove or disprove: If $span( v_1, v_2, ... , v_k) = V$, then $span(T(v_1), ... , T(v_k)...
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2answers
37 views

Linear map $L \neq O$ having trivial image only at $L^2=L \circ L$

From S.L Linear Algebra: Let $L:ℝ^2 \rightarrow ℝ^2$ be a linear map such that $L \neq O$ but $L^2=L \circ L=O$. Show that there exists a basis $\{A, B\}$ of $ℝ^2$ such that, $L(A)=B$ and $L(B)...
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2answers
30 views

Linear operator and inner product

Theorem: Let $V$ be an inner product finite space with an orthonormal basis $\mathcal B$. Let $L$ be an operator on $V$, and let $A = [L]_\mathcal{B}$, the matrix associate to $L$. Then the matrix ...
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0answers
22 views

Find a linear combination of vectors

$A,B,C$ and $D$ are collinear points. $B$ divides $AC$ in the ratio $2:5$ and $D$ divides $BC$ in the ratio $6:-1$. Express $\vec{OA}$ as a linear combination of $\vec{OB}$ and $\vec{OD}$ I do not ...
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2answers
37 views

Is $V = \{(a + 2b + 1, 2a-3b) | a,b\in\mathbb{R}\}$ a subspace of $\mathbb{R}^2$? Why or why not?

I am studying for a linear algebra final and going over the first exam. I just now retried this problem and I am making the same mistake I did the first time. I get that the set in NOT a subspace, but ...
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2answers
51 views

Is there a mathematical operation to ensure that two vectors are perfectly equal?

Let's said that I have a 3D matrix of integer of size 10x10x4, I want to know if along the 3rd dimension some vectors are perfectly similar (same number, same order). So I could compare one by one ...
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2answers
16 views

Claims regarding dimensions of vector space and subspaces

I have to check whether the following claims are true or false: Let $V$ be a vector space of finite dimension, and let $U, W$ be subspaces of $V$. 1) If $\dim V > \dim U + \dim W$, then $V \neq U ...
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1answer
29 views

Do we really mean “Cartesian Product of Vector Spaces” or is this just a naming convention?

While approaching tensors, I faced what is considered to be the cartesian product of 2 vector spaces. Now as much as I understood what this practically mean, I was quite concerned by its definition. ...
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0answers
24 views

adjoint operators in the vector space of real polynomials

This problem is about the space $V$ of real polynomials in the variables $x$ and $y$. If $f$ is a polynomial, $d_f$ will denote the operator $f(d/dx,d/dy)$ , and $d_f(g)$ will denote the result of ...
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2answers
37 views

Is the following a vector subspace over C? [closed]

$U$ is the set of matrices: $$\begin{pmatrix} a-2b & \bar c \\ b+c & 0 \end{pmatrix}$$ with $a,b,c \in \mathbb C$. Is $U$ a vector subspace over $\mathbb C$? Thank you!
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0answers
27 views

Proving that this is a vector space [closed]

I do not understand how to start with this proof. I have been taught about vector spaces and I understand how they work, yet I don't see where to start. Any help would be apreciated.
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0answers
38 views

Finding a subspace of dimension $3$ which does not intersect a rational subspace of dimension $2$

Some context. By rational subspace, I mean a subspace of $\mathbb R^5$ which admits a rational basis. In other words, a basis formed with vectors of $\mathbb Q^5$. For instance, the vector $v=(0,\pi,...
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1answer
26 views

Infinite linear combination

I don't know how to start the proof of the following statement: Let $V$ a real vector space with inner product $\langle\cdot,\cdot\rangle$. Consider a subset $C=\{v_j\}_J\subset V$ such that $\...
2
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1answer
52 views

Is the set of all vectors in the span of $\pmatrix{1 \\ 2}$ and $\pmatrix{2 \\ -1}$ a subspace or not?

I'm trying to figure out whether or not the set of all vectors in the span of $\pmatrix{1 \\ 2}$ and $\pmatrix{2 \\ -1}$ is a subspace or not (I know the answer to be yes, it is a subspace but I want ...
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2answers
38 views

How do determine whether a subset is a subspace?

So I have \begin{align*} A&= \{(x_1,x_2)'\in \mathbb{R}^2: \max (|x_1|,|x_2|) \leq 1\} \\ B &= \{(x_1,x_2,x_3,x_4)'\in \mathbb{R}^4:x_1⋅x_2=0\} \\ C&=\{(x_1,x_2,x_3)' \in \mathbb{R}^3 :...
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1answer
22 views

Is there a special relationship between a norm on a vector space V, and the operator norm $ \mathcal{L}(V, \mathbb{R)}$?

Let $T$ be a linear operator in $\mathcal{L}(V)$. An operator norm is denoted as $||T||$, where it is the smallest $M$, such that $||T(v)||$ $\le$ $M||v||$ for any $v \in V$. A norm on the vector ...
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1answer
32 views

Are they similar matrix

Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix} $ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix} $ are similar.Is this True/false ...
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2answers
26 views

Can any $n$ number of linearly independent vectors build the basis of a vector space on $\Bbb R^n$? [duplicate]

I'm trying to understand how many bases there can be in a vector space. According to my understanding so far, for example in the vector space $\Bbb R^2$ every pair of linearly independent vectors can ...
0
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2answers
15 views

prove that 2 vectors is linearly independent in the vector space of vectors of length 2 with entries of real-valued functions

I have the following vectors $V_1=(e^t,te^t), V_2=(1,t)$ now I want to prove that this two vectors are linearly independent in the vector space of vectors of length 2 with entries of real-valued ...
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2answers
29 views

Complement of a subspace

So I have this subspace $ U = span\{(1,1,1)^T,(0,1,-1)^T,(1,-1,3)^T\} $. I need to find the complement (W) of U such that $ \mathbb{R}^3 = U + W $ and $ U \bigcap W = 0$ . I'm not too sure how I'm ...
1
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1answer
51 views

Show that $(\mathbb{R},\mathbb{R},\odot,\oplus)$ is a vector space if $\odot$ and $\oplus$ are defined by:

Show that $(\mathbb{R},\mathbb{R},\odot,\oplus)$ is a vector space if $\odot$ and $\oplus$ are defined by $\alpha \odot x = \alpha^7 (x-3) + 3$ $x \oplus y = (\sqrt[7]{x-3} + \sqrt[7]{y-3})^7+3$ ...
2
votes
1answer
39 views

Bounds for the rank of the sum of two linear maps

The following is Exercise 3.15 from the German textbook Lineare Algebra by Hans-Joachim Kowalsky and Gerhard O. Michler: Let $\varphi$ and $\psi$ two linear maps from a finite-dimensional vector ...
0
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1answer
20 views

How do I solve this Matrice/Vectors question? [closed]

Give the matrix for the following linear operator: $A \vec{x} = (3,2,6) \times \vec{x}$, where $\vec{x}$ is any arbitrary vector.
0
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1answer
13 views

Is $ \max_{x\in\mathbb{R}^n} \{ f(x)+g(x) \} = \max_{x\in\mathbb{R}^n} f(x)+\max_{x\in\mathbb{R}^n} g(x) $ if $f$ and $g$ are affine in $\mathbb{R}$?

Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$. Is the following property true? $$ \max_{x\in\mathbb{R}^n} \{ f(x) + g(x) \} = \max_{x\in\mathbb{R}^n} f(x)...