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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2
votes
1answer
44 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 ...
7
votes
2answers
245 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}, {-^*}^* : ...
-3
votes
0answers
22 views

which book and document? [closed]

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?
0
votes
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 ...
1
vote
2answers
59 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 ...
0
votes
2answers
45 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 ...
0
votes
1answer
32 views

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 ...
0
votes
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?
1
vote
0answers
19 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}) ...
2
votes
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 ...
-1
votes
1answer
21 views

Vector spaces linear algebra problem [closed]

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 $Bx=0$
1
vote
2answers
115 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 ...
0
votes
1answer
27 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 ...
0
votes
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
votes
1answer
22 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 ...
0
votes
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 ...
1
vote
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
votes
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 ...
0
votes
2answers
17 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 \\ ...
1
vote
0answers
27 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 ...
1
vote
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 $\leq$ ...
0
votes
0answers
28 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
64 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 ...
0
votes
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?
1
vote
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 ...
-2
votes
1answer
38 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 $\mathbb C^2$ ?
0
votes
1answer
48 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
1
vote
0answers
28 views

Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
1
vote
1answer
23 views

Meaning of passing from a column to a row vector

When passing from column to row vectors in $K^n$ conceptually we're passing from a vector $(a_1,\ldots , a_n) \in K^n$ to it's associated linear functional defined by $ f(x_1,\ldots , x_n)=\sum_i a_i ...
1
vote
2answers
36 views

Explanation of additive identity and additive inverse in proving of vector space.

I know we have to prove these 10 properties to prove a set is a vector space. However, I don't understand how to prove numbers 4 and 5 on the list.
34
votes
6answers
3k views

What do mathematicians mean by “equipped”

I am a mathematical illiterate so I do not know what people mean when they say equipped. For example, I say that Hilbert space is a vector space equipped with a inner product. What does that ...
0
votes
0answers
35 views

Is the path integral the most general representation of the inverse of the Gradient operator?

Is the path integral the most general representation of the inverse of the Gradient operator? \begin{align} \boldsymbol{\nabla} \int_{\boldsymbol{x_0}}^{\boldsymbol{x}} \boldsymbol{F} \cdot ...
0
votes
2answers
24 views

How to find an orthogonal vector C in $C^3$ relative to two other (given) vectors?

$A = [2,1,-i]$ $B = [i, -1, 2i]$ I need to find a C that is orthogonal to A and B. I've tried taking AxB, but this does not work. I get the vector C = (i, 1-4i, -2-i). The problem is that ...
0
votes
0answers
12 views

calculating $y$ from $x^Ty=c_1$ with some constraints

It seems there are infinitely many solutions for this equation $x^Ty=c_1$, where $x$ (known) and $y$ (unknown) are vectors and $c_1$ is a positive value (calculating $y$ from the equation $u^Tv=x^Ty$ ...
1
vote
1answer
45 views

A question regarding Jacobi fields and families of geodesics

I'm trying to show that for any one-parameter family of geodesics $\gamma(s,t)$ (where $\gamma(s_0,t)$ is a geodesic for any constant $s_0 \in (-\epsilon, \epsilon)$) defined on a Riemannian manifold ...
0
votes
0answers
23 views

linear algebra, transformations [duplicate]

Let $V$ an $n$-dimensional vector space and $T$ a linear operator on $V$. Suppose that there is some positive integer $k$ such that $T^{k}=0$. Prove that. $T^{n}=0$
0
votes
1answer
48 views

Proving that S/I is a vector space

I'm given a polynomial ring $S=K[x_1,...,x_n]$ and $I$ is an ideal of $S$. I'm working on proving that the quotient ring $S/I$ is a vector spake over $K$. Since S is a ring, we already have some of ...
1
vote
1answer
20 views

Vector space of bidirectional sequences indexed by $\mathbb{Z}$

I am trying to claim the following: Consider the vector space of all $(x_i)_{i \in \mathbb{Z}}$ ($x_i \in \mathbb{C}$), over $\mathbb{C}$, where all $x_i$ are zero except finitely many of them. ...
1
vote
2answers
21 views

Find a linear transformation $F:\mathbb{R}^3\to\mathbb{R}^4$ that $\mbox{span}([1,1,2,1],[2,1,0,1])$

My book solves an exercise that asks to find a linear transformation such that its image is: $$\mbox{span}([1,1,2,1],[2,1,0,1])$$ The solution: Since $\mbox{dim Im}(F) = 2$, then $\mbox{Ker}(F) ...
2
votes
2answers
30 views

Is the norm of a projection of a vector along a subspace less than or equal to the norm of the vector iteself?

My question is, given a vector $x$ in a normed space, and any two subspaces with an intersection $0$ and whose direct sum is the whole vector space, is the norm of projection along one of the ...
-2
votes
1answer
34 views

calculating $y$ from the equation $u^Tv=x^Ty$ (all vectors)

Is it possible to calculate $y$ from the equation $u^Tv=x^Ty$ , where $x,y,u, v$ are all vectors? Assume $u,v,x$ are known and $y$ is unknown. Moreover, all the vectors have the same size, $n\times1$. ...
1
vote
1answer
29 views

When does $ \| Cx \|_{\mathbb{R}^2}^2 = \langle Cx, x\rangle$ hold?

When can one write the euclidean norm as an inner product? i.e. if : $$ \| Cx \|^2 = \sum^n_{i=1} x_i^2$$ then, when can we write the norm as follows: $$ \| Cx \|^2 = \langle Cx, x\rangle$$ I ...
3
votes
2answers
61 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
2
votes
1answer
36 views

Subspaces and polynomials

In a vector space $P_3$ of all real polynomials, where: $$H = \{p(t)=a_0 + a_1t + a_2t^2 + a_3t^3 \ \ | \ \ a_0 + a_2 = 0\ \ \text{and}\ \ \ a_1 + a_3 = 0\}$$ How do I show that $H$ is a subspace ...
1
vote
1answer
48 views

Expressing $C(x) = \tilde{x} = (\langle x,a_i \rangle )$ as a product of matrices in the form $Cx = \tilde{x}$

Le that $(a_i)^{n}_{i=1}$ be an orthonormal basis and $C(x)$ be a transformation defined as follows: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k ...
7
votes
2answers
58 views

For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem

Here it is a nice algebra problem I had some fun with Let $V$ be a vector space over $\mathbb R$ of finite dimension $\dim V = n$. Let $v = \{ v_1, \dots, v_n\}$ be a basis for $V$. Let $$S_{n,k} = ...
0
votes
1answer
42 views

Weird vector projection form

Let $C^0[1,3]$ be the $\Bbb R$-vector space equipped with the usual scalar product (by the integral ). Calculate the projection of the function $f(x)= 1/x$ onto the subspace $W = L\{x\}$ Well, my ...
0
votes
0answers
52 views

Vectors in tangent space to a manifold independent of coordinates

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator $X$ acting on some function ...
0
votes
1answer
23 views

Explanation of method for finding basis?

The method of row reduction allows you to find a basis for a subspace just by placing the vectors as rows of a matrix and then row reducting it. I've always took a linear combination of the vectors, ...