Tagged Questions

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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1answer
31 views

Vector Cross product - Rearranging issue

Given Data in question I have following relations in vector space$\begin{eqnarray}n_0^{'}(s)=-\kappa(s) \times n_0(s)\\n_1^{'}(s)=-\kappa(s) \times n_1(s)\\n_2^{'}(s)=-\kappa(s) \times ...
2
votes
2answers
52 views

Automorphism of algebraic closure $\overline{{\bf F}}_p$.

Problem : I want to give an concrete example of automorphism of $\overline{{\bf F}}_p$ which fixes ${\bf F}_p$, where $$\overline{{\bf F}}_p =\bigcup_{n\geq 1} {\bf F}_{p^n} $$ and ${\bf F}_{p^n} $ is ...
3
votes
1answer
61 views

Linearly Independent Vectors--Story Problem

Suppose we have a club with exactly $5$ students. Show, using vectors, that we cannot form $6$ groups so that every two groups share exactly $1$ student. So if we let $v_1 , \dots ,v_6$ be $6$ ...
2
votes
1answer
12 views

Infinite dimensional vector space and minimal generating subset

Let $V$ be an infinite dimensional vector space. Is it true that every generating subset of $V$ contains a basis of $V$? It is easy if $\dim V<\infty$, but I don't know how to proceed if $V$ is ...
1
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2answers
34 views

$\mathbb{C}$ Forms a Vector Space Over $\mathbb{R}^2$ Proof Question

In my Mathematical Techniques course we've been talking about vector spaces, bases, etc. There is one problem however that I cannot get my head around and that is to prove that $\mathbb{C}$ can be ...
2
votes
1answer
29 views

Base of Continuous functions

is there a method to determine one of the infinite bases of the vector space of continuous function over an interval [a,b] (a If the question is not "well asked" : How two functions in this space are ...
1
vote
2answers
26 views

Show that $S_0$ is a supbspace of $S$

"Let $S$ denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by $a\{b_n\} = \{ab_n\}$ (where $a$ is a scalar) and $\{b_n\} + \{c_n\} = \{b_n ...
0
votes
2answers
56 views

To prove $\det (xy^t)=0$ [duplicate]

Let $x,y$ be arbitrary non-zero column vectors in $\mathbb R^n$ , then how do we prove that $\det (xy^t)=0$ ?
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0answers
14 views

What is an optimal order for integer vectors for minimization of the total distances?

I want to find an optimal order for a number of vectors (or a permutation of vectors) to minimize the sum of distances regarding to the following norm: (this norm is based on the distance on a cycle ...
0
votes
3answers
32 views

Quick question about subspaces, bases and linear independence.

Let $X$ be an $n-$dimensional vector space and $Z$ be an $(n-1)$-dimensional subspace of $X$. If $\{e_1, ..., e_{n-1}\}$ is a basis for $Z$, is it true that I can always find a vector $e_n \in X$ ...
0
votes
0answers
37 views

Prove that $\dim(V)$ is an even number

Let $V$ vector space so that $\dim(V)=n$ and let $T:V\to V$ a linear transformation so that $\mathrm{Im}(T)=N(T)$. Prove that $\dim(V)$ is an even number I have no idea how to star the problem. Can ...
0
votes
1answer
27 views

Prove that every element of $V$ can be expressed as $w+cv_0$ for some $w\in \ker(T)$ and $c\in \mathbb R$

Let $V$ be a vector space over $\mathbb R$ and let $T:V\to \mathbb R$ a linear transformation. Suppose $\ker(T)\neq V$ and let $v_0\in V$ so that $v_0\notin \ker(T)$. Prove that every element of $V$ ...
0
votes
0answers
35 views

$3D$ surfaces multivariable Calculus

A surface is constructed as follows: First a curve $(0, y, −((y − 1)^2)((y + 1)^2))$ is drawn in the yz–plane. Then a parabola $(u, u^2)$ is drawn in the uv–plane. Finally, in each plane y = b, a copy ...
0
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0answers
19 views

How can i get a vector of fixed magnitude in a particular direction?

I am programming something and i am stuck with this problem related to mathematics. I am not so good in maths. I have Vector A and Vector B. I know that to get the direction from A to B, I have to ...
0
votes
1answer
28 views

Basis of the row-space of a matrix with non-negative entries.

Consider a matrix $A \in \mathbb{R}^{n \times m}$ such that all entries are non-negative. Denote the rank of $A$ as $k$. I am mostly interested in cases where $k \ll n$, but this probably isn't ...
-3
votes
2answers
38 views

How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
0
votes
1answer
13 views

dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
0
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0answers
11 views

Tangential and normal components of acceleration acting on a dropped bomb

The original question is this: A plane flying at an altitude of 34,000 feet at a speed of 510 miles per hour releases a bomb. Find the tangential and normal components of acceleration acting on ...
0
votes
0answers
25 views

Determining whether $\mathbb R^+$ is a vector space under given actions

While $\mathbb R^+=V$ ($\mathbb R$ is the group of all real positive numbers), we define the following actions: $\forall a, b \in V$, $a \oplus b = a \cdot b$ $\forall a \in V$, $\forall ...
0
votes
1answer
19 views

Proving properties of orthogonal subspaces

If $M$ and $N$ are subspaces of $\mathbb{R}^n$, show that $$(M+N)^\bot=M^\bot \cap N^\bot$$ and $$(M \cap N)^\bot = M^\bot +N^\bot $$ I found this previous question, but I cannot seem to make sense ...
1
vote
1answer
77 views

Quotient Space (W-Affine Subspaces) “Proof” Verification.

Let $(V, K)$ be a vector space and $W ⊂ V$ a subspace. A subset $S ⊂ V$ is called a $W$-affine subspace of $V$ if the following holds $∀s, s ∈ S, s − s ∈ W$ and $∀s ∈ S, ∀w ∈ W, s + w ∈ S$. ...
5
votes
2answers
122 views

Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
0
votes
2answers
89 views

Isomorphism of all Linear Transformation

my work book has a questions that asked us to prove something, however the answer was not provided. The question states that: Let $V$ and $W$ be vector spaces over $F$ where the $dim(V) = n, dim(W) = ...
0
votes
1answer
45 views

Show that V is a subspace by expressing it as the span of a set of vectors

What exactly is this question asking me to do? I think the use of the set notation has thrown me off a bit. Any help is appreciated.
-1
votes
0answers
20 views

Basis span of space necessary to be orthogonal?

Q1 If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal? (From several users comments, the answer is the basis vector can be no mutually ...
0
votes
2answers
37 views

Finding a basis for a subspace in $\;\Bbb R^4\;$

I know this might be a really simple question to ask but I just don't understand how to obtain the answer to this question. I've tried to understand subspaces (and even the difference between a space ...
0
votes
1answer
33 views

Are the electric and magnetic fields functions on R^4?

Are the electric and magnetic fields functions from $\mathbb{R}^4$ to $\mathbb{R}^4$ (where $\mathbb{R}^4$ is then interpreted as space-time) or do we consider them to be functions from $\mathbb{R}^3$ ...
0
votes
3answers
33 views

show that dim(L,W) = mn

There are two finitely dimension vector spaces $V$ and $W$. Dimensions are $n$ and $m$ respectively. $$L(V,W)=\{T:V\rightarrow W \;|\; T \;\text{is linear}\}$$ $L(V,W)$ is a vector space with ...
0
votes
1answer
18 views

What can I say about the dimension of all real functions?

If I have a vector space of all real functions And S is all real functions with no constant term. then S is the subspace of V. Then, What can I say about the dimension of S? V has infinite ...
1
vote
1answer
41 views

Why is the infinite dimensional vector space with only finitely many nonvanishing components incomplete?

Define a complex vector space $V$ such that any element $\{a_i\}=(a_1,a_2,\dots)\in V$ has only finitely many components $a_i\ne 0$. The inner product is defined as $$(\{a_i\},\{b_j\})=\sum_i^\infty ...
0
votes
0answers
38 views

Rank Nullity Theorem for Infinite dimensional vector spaces

Rank nullity theorem can be extended for infinite dimensional vector space.Can someone help me to complete my proof.I think this idea will work. Rank Nullity Theorem states that if $T$ is a linear ...
0
votes
1answer
19 views

Proving the 0 Vector and the Set of Eigenvectors of a Linear Map are a Subspace

Given the map T∈ L(V) where L(V) is the set of all linear maps from V to V. I'm wondering whether it can be proven that the set of {the 0 vector and all the eigenvectors of T} can be shown to be a ...
0
votes
1answer
29 views

Easy question about vector spaces

Suppose $F$ is a (added later: finite-dimensional) vector space over $K$ and $K'$ is a subfield of $K$. If $\dim_K F = \dim_{K'} F$, then how does one prove that $K=K'$? Somehow I can't quite show ...
0
votes
0answers
39 views

Existence of a basis in constructive vector spaces

As I was trying to review forgotten knowledge on Vector Spaces in wikipedia, I read that the existence of a basis follows from Zorn lemma, hence equivalently from the axiom of choice. Actually, the ...
0
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0answers
58 views

Find 2 unit vectors that make an angle of $60^\text{o}$ with $\vec v=\langle 3,4 \rangle$

Find 2 unit vectors that make an angle of $60^\text{o}$ with $\vec v=\langle 3,4 \rangle$. My working: $$\cos{60^\text{o}}=\frac{1}{2}= \frac{\langle u_1,u_2\rangle\cdot\langle ...
0
votes
1answer
39 views

3D plane rotation about a line

In three dimensional space we have a plane and a line. These can be oriented in any way. The plane is rotated about the line by n degrees, meaning that originally the position of the plane is fixed to ...
1
vote
1answer
58 views

Is there a difference between $a \cdot a^T$ and $a^2$?

The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a ...
0
votes
2answers
40 views

Find a normal vector onto the line

How can I find normal vector on the given line. For example if I have a line $3x - 5y = 1$, what would be the normal vector of this line? I am not sure whether it's useful or not, but we have one more ...
1
vote
1answer
40 views

Find a basis for a subspace (working included)

I have been working on this question and I am not too sure if it is correct or not. Any help would be appreciated. Question (in picture format): http://i.imgur.com/E4MhH99.png My working: The first ...
-1
votes
1answer
25 views

Proof that R2 belongs to (a+b, b) [duplicate]

I am aware that the vector (a+b, b) belongs to R2 for a,b being real numbers. Also, I am aware that R2 belongs to (a+b,b), but I am not sure how to prove it. R2 is defined as (x1, x2), x1,x2 are real ...
1
vote
2answers
58 views

Proof that $(a+b, a)$ belongs to $\mathbb{R}^2$

I am aware that the vector $(a+b, a)$ such that $a$, $b$ are real numbers belongs to $\mathbb{R}^2$, which is defined by any vectors $(x_1, x_2)$ such that $x_1, x_2$ are real numbers. Is there a way ...
0
votes
2answers
35 views

$v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$

Let $E,F$ two $\mathbb{K}$ vector spaces, $u\in\mathcal{L}(E,F)$. a) Show that there exists $v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$ b) Can we additionally have $v\circ u\circ v=v$ ? ...
0
votes
1answer
34 views

Understanding the operator of differentiation on the vector space of polynomials

I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the ...
1
vote
2answers
46 views

What is wrong with this proof that if $V = U_1 \oplus W$ and if $V = U_2 \oplus W$, then $U_1 = U_2$?

Claim: Let $U_1, U_2$ and $W$ be subspaces of a vector space $V$. Suppose $V = U_1 \oplus W$ and $V = U_2 \oplus W$. Then $U_1 = U_2$. "Proof" Let $v \in V$. Then $\exists \space u_1 \in U_1 $ ...
0
votes
2answers
29 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
2
votes
1answer
38 views

Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
-3
votes
0answers
23 views

dimension of intersection of two subspaces [closed]

$w_1=${$(0,x_2,x_3,x_4,x_5)\hspace{0.1in} | \hspace{0.1in} \forall x_i \in \mathbb{R} \hspace{0.1in} i = 2,3,4,5$ } & $w_2=${$(x_1,0,x_3,x_4,x_5)\hspace{0.1in} \vert \hspace{0.1in} \forall x_i ...
0
votes
2answers
36 views

What does 'dimension' strictly mean?

Ask a simple question but confusing me. Case 1. Take an Eucildean space R^3 for example. R^2 is one of its subspce with bases [1,0] and [0,1], and the dimension of this subspace is 2. So for example ...
0
votes
2answers
55 views

Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
2
votes
1answer
20 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...