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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17
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1answer
1k views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
4
votes
2answers
435 views

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
40
votes
4answers
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Why are vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
11
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7answers
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A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
26
votes
2answers
3k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
4
votes
1answer
222 views

Is it possible to construct a quasi-vectorial space without an identity element?

I mean if there is any construction that satisfies all the conditions for an vectorial space except it lacks an identity element? This questions was posed to me by a classmate last semester and I have ...
14
votes
8answers
22k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
11
votes
4answers
4k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
16
votes
3answers
2k views

Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an ...
5
votes
2answers
815 views

Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
2
votes
3answers
519 views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be ...
1
vote
0answers
52 views

Proof by contradiction: $E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$

I must proof the following: Prop.: Let $E_1,E_2$ two vector subspace of $V$ then $$E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$$ Proof: I must show $$1)E_1+E_2\doteq E_1 \oplus ...
21
votes
6answers
6k views

Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal ...
11
votes
1answer
2k views

Transpose of a linear mapping

There seems to be two kinds of transposes of a linear mapping: If $f: V→W$ is a linear map between vector spaces $V$ and $W$ with nondegenerate bilinear forms, we define the transpose of $f$ ...
19
votes
3answers
873 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
9
votes
2answers
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How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
3
votes
2answers
194 views

A union of two subspaces not equal to the vector space. [duplicate]

Let $L,M$ two subspaces of the vector space, $V$ such that both $L,M \ne V$. Prove: $L\cup M \ne V$. I think this is a case of a proof by contradiction. Lets assume $L \cup M = V$. Hence, ...
19
votes
2answers
5k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
14
votes
2answers
3k views

Understanding isomorphic equivalences of tensor product

I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following ...
9
votes
5answers
3k views

Covectors and Vectors

I have a general question about vector/covectors: Background. A vector (for our purposes) is a physical object in each basis of $\mathbb{R}^3$ represented by three numbers such that these numbers ...
14
votes
1answer
660 views

Vector Spaces and AC

I know that the proof that every vector space has a basis uses the Axiom of Choice, or Zorn's Lemma. If we consider an axiom system without the Axiom of Choice, are there vector spaces that provably ...
7
votes
1answer
18k views

Using the Determinant to verify Linear Independence, Span and Basis

Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a ...
5
votes
2answers
604 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
5
votes
3answers
346 views

How to think of a function as a vector?

In order to apply the ideas of vector spaces to functions, the text I have (Wavelets for Computer Graphics: Theory and Applications by Stollnitz, DeRose and Salesin) conveniently says Since ...
4
votes
1answer
18k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
3
votes
3answers
1k views

Interior of a Subspace

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
14
votes
3answers
8k views

How to find basis for intersection of two vector spaces

What is the general way of finding the basis for intersection of two vector spaces? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), ...
2
votes
3answers
430 views

How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?

I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.
3
votes
0answers
162 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
3
votes
2answers
249 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
3
votes
2answers
785 views

Scalar Product for Vector Space of Monomial Symmetric Functions

Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $$ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $$ Since ...
1
vote
1answer
91 views

Why is class of one-dimensional vector spaces not axiomatizable?

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar $\alpha$ of R. There are ...
1
vote
2answers
510 views

Is this Vector operation defined? Does it have a name?

Let's say I have 2 vectors: [a, b, c] [x, y, z] And I need to do an operation like the following for a computer program: ...
0
votes
2answers
53 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
0
votes
1answer
149 views

Functions $ \cos(2x)$, $\sin(2ax)$, $1$ independent and dependent

For which value(s) of $a$ are the functions $\cos(2x)$, $\sin(2ax)$, $1$ independent over the real numbers? For which $a$ are they dependent? I thought maybe to equate each (with the use of ...
13
votes
3answers
527 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
8
votes
2answers
2k views

Matrix of Infinite Dimension

Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces. But if one or all of the vector spaces is infinite dimensional, is the ...
6
votes
2answers
823 views

While proving that every vector space has a basis, why are only finite linear combinations used in the proof?

Statement: Every vector space has a basis Standard Proof:It is observed that a maximal linearly independent set is a basis. Let $\mathscr{Y}$ be a chain of linearly independent subsets of a ...
7
votes
2answers
755 views

$k[x]$-module and cyclic module over a finite dimensional vector space

Given a finite dimensional vector space $V$ over a field $k$ and a linear transformation $T: V \rightarrow V$ we can make $V$ a $k[x]$-module via the map: $$(a_{0}+a_{1}x+\cdots+a_{n}x^{n}) \cdot v ...
4
votes
1answer
366 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product ...
4
votes
4answers
1k views

What are some alternative definitions of vector addition and scalar multiplication?

While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that ...
3
votes
2answers
185 views

$2\times 2 $ matrices over $\mathbb{C}$ that satisfy $A^3=A$

Let $A$ be a $2\times 2$ matrix with complex entries. What would be the number of $2\times 2$ matrices $A$ that satisfies $A^{3} = A$. Question was are they infinite? If it is $3\times 3$ matrix then ...
4
votes
1answer
352 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the ...
4
votes
1answer
253 views

Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose ...
4
votes
1answer
405 views

Splicing together Short Exact Sequences

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of ...
2
votes
1answer
280 views

Obtaining Least square adjusted single line by intersecting many 3D planes

I am working with many 3D planes and looking for a Least square solution for below case. IF I am having many number of 3D planes knowing only one point and the normal vector (for eg. O1 and N1), ...
11
votes
3answers
416 views

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
5
votes
2answers
73 views

Necessary condition for have same rank

Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix. Prove that $P$ and $Q$ have the same rank. Some help with this please , happy year and thanks.
4
votes
3answers
100 views

For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?

I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you ...
3
votes
1answer
2k views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...