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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The set of differentiable functions such that $f'(2)=b$ is a linear subspace if and only if $b=0$

The set of differentiable real-valued functions on (0,3) such that $f'(2)=b$ is a subspace of $(0,3)\to \mathbb R$ if and only if $b=0$ ($(0,3)\to \mathbb R$ denotes the set of functions from $(0,3)$ ...
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2answers
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Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
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2answers
22 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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3answers
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I have difficulty understanding functions forming vector space.

I have knowledge of basic linear algebra, so I can understand the finite vector space as linear combinations of vectors of $R^n$. However, when it comes function as vector and functions form a ...
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3answers
18 views

Find all linear operators such that $F^2 = F$ and $F(x,y) = (ax,bx+cy)$

I need to find all linear operators that match $F^2 = F$ and $F(x,y) = (ax,bx+cy)$ *where $F^2$ means $F$ composed with itself. So what I did: $F(x,y) = (ax,bx+cy)\implies F(F(x,y)) = ...
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4answers
140 views

Definition of basis

There are something that I am not quite sure about the definition of basis. Let $V$ be a vector space over $K$, then the definition of basis says the vectors $v_1,...v_n$ form a basis of $V$ if they ...
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2answers
60 views

$V$ is finite dimentional over field $K\iff$ field extension $L/K$ is finite

Let $L/K$ be a field extension and $V$ a non-zero vector space over $L$. Prove that: $V$ is finite dimensional over $K\iff V$ is finite dimensional over $L$ and $[L:K]<\infty$ for the first ...
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1answer
35 views

Find a basis for $\mathbb{R} ^5$ containing the given vectors

Find a basis of $\mathbb{R}^5$ that contains the vectors $(1,-1,1,-1,0)$, $(-1,-1,1,-1,0)$ , $(-1,1,1,-1,0)$. I think I need to find two more vectors so that the five vectors are all linearly ...
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0answers
21 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
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1answer
19 views

Finding same-vectors that have same coordinates in two different basis

I have two different vector basis: Default: $\{e_1,e_2,e_3\} = \{(1,0,0);(0,1,0);(0,0,1)\}$ Special basis: $\{e'_1,e'_2,e'_3\} = \{(1,1,1);(1,0,1);(0,2,1)\}$ My question is: How do I find which ...
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2answers
41 views

Let $F$ be a linear operator such that $F^2 - F + I = 0$, show that $F$ is invertible and $F^{-1} = I - F$

I didn't understand this exercise. I tried working with $$F^2 - F + I = 0\implies (F-I)(F) + I =0$$ but I really don't understand how to prove $F$ is invertible neither find the inverse. Any hints? ...
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1answer
19 views

Verifying if $F$ such that $F(1,0) = (2,5)$ and $F(0,1) = (3,4)$ is an automorphism

What I did: $$(x,y) = x(1,0) + y(0,1)\implies\\F(x,y) = xF(1,0) + yF(0,1)\implies\\F(x,y) = x(2,5) + y(3,4) = (2x+3y, 5x+4y)$$ I need to verify if $G = I + F$ is na automorphism. So: $$G = I + F = ...
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1answer
11 views

Tetrahedron in vector space: Finding a vector connecting two points

Edited to add: The tetrahedron is not necessarily a regular one. First off, the point $M$ is the centre of gravity for this tetrahedron. I have a base $\{e_1,e_2,e_3\} = ...
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0answers
8 views

Curl of a 2D Spherical Field [on hold]

I have a (r,θ) 2D spherical polar field and I want to find its curl. How would I go about doing this? Thanks in advance.
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2answers
14 views

Rank of a matrix from a 5 X 7 matrix with a basis of 3 vectors

The question in my book is as follows: If the subspace of all solutions of Ax=0 has a basis consisting of thee vectors and if A is a 5 x 7 matrix, what is the rank of A? Now i thought because ...
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1answer
137 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if ...
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1answer
89 views

Estimating rank and nullity of the composition of linear maps

Let $T\colon U\to V$, $R\colon V\to W$ be linear maps between finite dimensional spaces $U$, $V$, $W$, and let dim$(V)=n$. Prove that $\dim\, \ker(RT)\le \dim\, \ker(R)+\dim\, \ker(T)$, ...
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1answer
27 views

Is this set of functions a vector space?

I'm starting to learn linear algebra am an learning what is and what is not a vector space. I'm trying to figure out if the following set of functions is a vector space: {f : R → R | f(3) = 0} I ...
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1answer
41 views

Are functions infinite dimensional vectors? [on hold]

Are functions infinite dimensional vectors? There are a few sources on the internet that makes this claim, but they do not cite any sources which makes me feel like they are just using it as an ...
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1answer
29 views

$cu=0\iff u=0\vee c=0$

$c$ is a constant in R or C $u$ is in a vector space $cu=\mathbf 0\iff u=\mathbf 0\vee c=0$ First I tried to show the two implications $cu=\mathbf0\Leftarrow u=\mathbf0\vee c=0$ and ...
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2answers
56 views

Is the empty set a vector space?

I think the empty set satisfies all of the axioms of a vector space except the one about the existence of an additive identity. Is this right?
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34 views

Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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1answer
26 views

Conceptual question about the extended real line and being a vector space.

Last time I was chatting with a professor online on a public IRC, this is a transcript: #Professor (16:20:12): So what was your question? #Me (16:20:22): I feel stupid.... #Professor ...
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1answer
41 views

Is a subset a vector space if it is just one vector?

I'm beginning to learn linear algebra and wanted to know if a subset of a vector space is a vector space if there is only one vector. For example, $$V = \{(x, y, z) \in \mathbb{R}^3| x^2 + y^2 + z^2 ...
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2answers
241 views

Finite-dimensional space naturally isomorphic to its double dual?

The example that a finite vector space is naturally isomorphic to its double dual seems to be the canonical example of natural isomorphisms. Concretely, there are two functors $\mathsf{Id}, {-^*}^* : ...
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0answers
21 views

which book and document? [on hold]

I want to study about weak topology and weak star topology. So, what can I read books or documents? With 'Functional analysis', which books are good to study?
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1answer
20 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
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2answers
57 views

Prove linear independence of a set $\{\mathbf{x}-\mathbf{x_1},\ldots,\mathbf{x}-\mathbf{x_n}\}$

Let $V$ be a vector space and suppose that $\{\mathbf{x_1},\ldots,\mathbf{x_n\}}$ is a linearly independent subset of $V$. If $\mathbf{x} = \sum_{i=1}^n c_i\mathbf{x_i}$ where each $c_i \in ...
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2answers
44 views

Prove that a Polynomial ring is a vector space

so I shall prove that a Polynomial ring [K] is a vector space. How do I do that? I was thinking of just going down all axioms one by one.. but I don't really know how to prove them for a polynomial ...
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1answer
21 views
+50

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
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0answers
13 views

Expressing a vector in terms of an Arbitrary Vector?

Ive been working on vector questions and this one seems to have gotten me stuck. Im unsure on how to express v in terms of i and j such as: (v . i)i + (v . j)j Can somebody provide some help?
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17 views

How to solve 3D vector equations symbolically?

I'm trying to solve the following system of equations for $\vec{X}$: $$ \vec{A} \cdot \vec{X}=d_1 $$ $$ \vec{B} \cdot \vec{X}=d_2 $$ $$ (\vec{A} \times \vec{B}) \cdot \vec{X}=(\vec{A} \times \vec{B}) ...
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1answer
37 views

Points with each pair having distance in range

What is the maximum number of points can be placed on a plane such that the distance between any two is in some range? Specifically I'm interested in the range $[4,5]$, although I'm interested also in ...
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1answer
21 views

Vector spaces linear algebra problem [on hold]

Given: A invertible matrix $(n\space \times \space n)$ B matrix $(n\space \times \space m)$ prove that the Solution space of the system $ABx=0$ is equal to $Bx=0$
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1answer
71 views
+50

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
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1answer
25 views

trying to prove that a span is a basis for another set

question I know that the correct way to solve this question is to prove that s is a subset of B and B is a subset of S so that B is a basis for S. I managed to prove that s is a subest of span B, but ...
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1answer
28 views

Rank of linear transformations from vector space to same vector space

I have a question about a homework question so don't expect a full solution. Just if someone could tell me how I should approach this question. I'm not really sure what kind of manipulation is ...
0
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1answer
21 views

Get tangent vector from point to sphere vector

I have a sphere that has it's center at $A$ and lets say a radius of 1. Then I have a point $C$, some units away from the sphere in an arbitrary direction. I can easily calculate a vector, lets say ...
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1answer
16 views

Finding base for a set of vectors

Given these sets of vectors: $$ T=\{(2,1,-1),(1,0,-1),(5,1,-4)\} $$ $$ S=\{(1,2,1),(1,1,2),(3,4,5)\} $$ 1) Find a base for the subspaces: $Sp(S)$, $Sp(T)$, $Sp(S\cup T)$ 2) Describe the vectors ...
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1answer
18 views

Clarification on definition of a basis

Quick question; lets say that $S$ is a basis of $V$. I understand that this means all vectors in $S$ are linearly independent, and that every vector in $V$ is an element of $\text{span} \ S$. Is it ...
2
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2answers
38 views

linear algebra problem in matrices

I have no idea how to approch this, any help will be greatly appreciated: Given: Matrix A of order $(k\times n)$ Matrix B of order $(n\times k)$ with $k\neq n$, prove that its not possible for ...
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2answers
16 views

Find the dim of the solutions for $Ax=0$

Let $A$ be a matrix: $$ A=\begin{pmatrix} 1 & 1 & -5 & -6 & 1 \\ 2 & 1 & -7 & -7 & 1 \\ 1 & 2 & -8 & -11 & 5 \\ ...
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0answers
22 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
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1answer
29 views

Linear space and cardinals

Let $V$ be a linear space over a field $K$. If $L, S\subseteq V$ such that $L$ is a linear independent set with the property that $L\subseteq \mathrm{Span}(S)$, prove that: $|L|\leq |S|$, where $\leq$ ...
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0answers
27 views

L0 norm, L1 norm and L2 norm

For vector $\boldsymbol{x},\boldsymbol{y} \in \mathcal{R}^{n}$, if \begin{equation} \| \boldsymbol{x} \|_0 = \| \boldsymbol{y} \|_0 \end{equation} What relationship will $\| \boldsymbol{x} \|_1$ and ...
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0answers
25 views

Drawing a Parametric Equation from the intersection of surfaces

I need help with the second part of this problem. Show that any point on $$x^{2}+y^{2} = z^{2}$$ can be written in the form $$(zcos\theta ,zsin\theta ,z)$$ for some $\theta$. Use this to find a ...
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1answer
60 views

How could we prove that it is not a spanning set.

Consider the space $\mathbb{R}$ as a linear space over the field $\mathbb{Q}$ of rational numbers. For any transcendental number x the set {1, $x$, $x^2$, $x^3$,......} is linearly independent. How ...
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1answer
29 views

What is a comoving basis?

I have read that the tangent vector, principal normal vector and binormal vector consistute a comoving orthogonal basis. But in this context what does comoving mean?
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1answer
48 views

Are 'vectors' vectors?

Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a ...
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1answer
37 views

Can there exist a linear operator $T : \mathbb C^2 \to \mathbb C^2$ such that $\langle T(v) , v \rangle =0$ ? [closed]

Can there exist a linear operator $T : \mathbb C^2 \to \mathbb C^2$ such that $\langle T(v) , v \rangle =0$ , where $\langle ., .\rangle$ is the usual inner product over $\mathbb C^2$ ?