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 we ...
24
votes
3answers
2k views
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 ...
4
votes
1answer
177 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 ...
16
votes
2answers
1k views
Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)
I am trying to understand the differences between
$$
\begin{array}{|l|l|l|}
\textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline
\text{metric}& \text{metric ...
3
votes
2answers
294 views
If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$
If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
9
votes
1answer
558 views
Transpose of a linear mapping
There seems to be two kinds of transposes of a linear mapping:
If $f: V→W$ is a linear map between vector spaces $V$ and $W$ with
nondegenerate bilinear forms, we define the transpose of $f$
...
9
votes
2answers
660 views
How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
I want to prove the following theorem (no idea whether it has a name):
Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
6
votes
2answers
918 views
Understanding isomorphic equivalences of tensor product
I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following ...
12
votes
1answer
394 views
Cardinality of a Hamel basis
What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
11
votes
3answers
730 views
Inner Product Spaces over Finite Fields
Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$.
I want to know what happens if we try to define them over some finite field. Here's an ...
4
votes
2answers
468 views
Power-reduction formula
According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following:
$$
\cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
3
votes
2answers
502 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
2answers
279 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:
...
5
votes
5answers
1k 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 ...
9
votes
7answers
488 views
Simple fact on linear operators
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 ...
4
votes
2answers
952 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 ...
4
votes
4answers
675 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 ...
4
votes
1answer
84 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
4answers
92 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
3
votes
1answer
64 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 ...
9
votes
1answer
238 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 ...
3
votes
2answers
111 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 ...
2
votes
1answer
47 views
Proving linear independence
Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy:
$$
Av_1 = v_1 \\
Av_2 = 2v_2 \\
Av_3 = 3v_3 $$
Prove that $\{v_1, v_2, v_3\}$ is ...
1
vote
2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
1
vote
2answers
66 views
Calculating new vector positions
I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
1
vote
3answers
328 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 ...
1
vote
1answer
250 views
Could intersection of a subspace with its complement be non empty.
If that is possible could you please correct my understanding about complement of a subspace.
From what i recall from set theory. A complement of a set B is the set U - B where U is the universal ...
0
votes
4answers
167 views
Basis of a $2 \times 2$ matrix with trace $0$
I have a question that I do not understand and it goes like this:
Find a basis for the set $W$ of all matrices A in $M_{2\times2}$ with trace $0$: i.e. all matrices
$$
\begin{pmatrix}
a & b\\
c ...
0
votes
2answers
815 views
Calculate intersection of vector subspace by using gauss-algorithm
There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$
My idea was to calculate the Intersection of those two ...
0
votes
1answer
158 views
How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $? [duplicate]
Possible Duplicate:
Lower bound involving the rank of the composition of linear transformations
The following question is about a lower bound on the rank of a composition of functions given ...
0
votes
2answers
138 views
Lower bound involving the rank of the composition of linear transformations
The following question is about a lower bound on the rank of a composition of functions given as a simple expression for the two terms of the sum involved in the inequality.
Consider ...
25
votes
0answers
621 views
What is the solution to Nash's problem presented in “A Beautiful Mind”?
I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
23
votes
4answers
1k views
Given two basis sets for a finite Hilbert space, does an unbiased vector exist?
Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that:
$$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
12
votes
2answers
2k 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 ...
6
votes
1answer
575 views
Vector, Hilbert, Banach, Sobolev spaces
Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
7
votes
3answers
4k 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 ...
5
votes
3answers
5k views
How to tell if a set of vectors spans a space?
I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
5
votes
3answers
379 views
Why do we use n-dimensional spaces?
On mathoverflow, Terry Tao says the following:
For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, ...
9
votes
3answers
779 views
Question about basis and finite dimensional vector space
I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5)
I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
6
votes
2answers
198 views
Can a basis for a vector space be made up of matrices instead of vectors?
I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
5
votes
3answers
226 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
144 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 ...
3
votes
5answers
220 views
Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$
Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of
$V$. Assume that
\begin{eqnarray}
\sum^{k}_{i=1} \dim(V_i) > n(k-1).
\end{eqnarray}
To show that ...
3
votes
2answers
344 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 ...
3
votes
3answers
145 views
Possible proof for the relation involving matrix trace
Suppose a diagonal matrix $D\in\mathbb{R}^{n\times n}$ is given, with all its entries $d_{ii}\geq0$, for all $i$. Is it possible to prove
...
2
votes
1answer
634 views
Relation between cross-product and outer product
If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way?
A quick search reveals that ...
9
votes
4answers
1k 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 ...
8
votes
6answers
610 views
Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}[0,1]$
How do i show that $f_{1}(x)=1$, $f_{2}(x)=e^{x}$ and $f_{3}(x)=\sin{x}$ are linearly independent, as elements of the vector space, of continuous functions $\mathcal{C}[0,1]$.
So for showing these ...
7
votes
3answers
3k 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), ...
5
votes
4answers
95 views
Vector dimension of a set of functions
Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$.
It is easy to prove that $|S| \leq ...
4
votes
3answers
89 views
How do you construct the quaternion and the multiplication rules, like Hamilton did?
So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
