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 ...
26
votes
0answers
686 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 ...
11
votes
0answers
90 views
Combinatorics in finite vector space
Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$.
Let $k$, $a$ and $b$ be non-negative integers.
Determine the number of subspaces $K$ of $V$ ...
4
votes
0answers
106 views
Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?
An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
4
votes
0answers
45 views
In stating that the union of vector subspaces is a subspace iff they are ordered, why require $F$ finite?
On the bottom of page 38 of Roman's Advanced Linear Algebra is written the following (here $V$ is a vector space over the field $F$ and $\mathcal{S}(V)$ is the set of linear subspaces of $V$):
"...if ...
4
votes
0answers
91 views
Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form?
I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question.
Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
4
votes
0answers
63 views
Extensions of finite-rank operators
Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the ...
3
votes
0answers
36 views
find the dimension of $W.$
Let $W=\{p(B):p \text{ is a polynomial with real coefficients}\},$ where $B=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}.$ Then find the dimension of $W.$
I have shown ...
3
votes
0answers
211 views
Dimension of Vector Space (Polynomial)
I was asked by a friend to: "Find the dimension of the vector space consisting of all polynomials in $n$-variables of degree at most $k$".Now, my response to him was that since the basis consists of ...
3
votes
0answers
100 views
Divergence Theorem to prove equality of integrals
I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem.
Prove that
$$\int_\Omega u \cdot \Delta v\, ...
3
votes
0answers
40 views
Symmetrizing a sequence of vectors
Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
3
votes
0answers
445 views
Three-dimensional vectors and force systems
Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below
As shown, a system of cables suspends a crate weighing W = 350 .
(Part C 1 figure) ...
3
votes
0answers
82 views
Self-absorbing subsets in a vector space
From planetmath
Let $V$ be a vector space over a field $F$ equipped with a
non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two
subsets of $V$. Then $A$ is said to absorb ...
3
votes
0answers
43 views
A question about pyramids (polytopes)
Let's use the following definition of a face:
A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
2
votes
0answers
51 views
Transposition of Composition is Reversed Composition of Transpositions
I'm trying to show that $(UT)^*=T^*U^*$.
Here is my effort:
Consider the following data:
\begin{array}{lcl}
T:V\rightarrow W & \leadsto & T^*:W^*\rightarrow V^* \\
U:W\rightarrow Z & ...
2
votes
0answers
35 views
How prove this $\frac{|x-z|}{|x-y|}=1+\frac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$
prove that
$$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$
for $|x|\longrightarrow \infty$
where
$$\hat{x}=\dfrac{x}{|x|}$$
This problem from book,following is my idea:
...
2
votes
0answers
51 views
Is there a name for this type of matrix?
I am preparing to publish an academic article on computational efficiency and image processing. In my work, I have come across what I can best describe as a non-square skew (symmetric or repeating) ...
2
votes
0answers
46 views
Dot products of three or more vectors
Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ ...
2
votes
0answers
57 views
Simple linear operator?
From Wikipedia
a linear operator T on a finite-dimensional vector space is semi-simple if every T-invariant subspace has a complementary T-invariant subspace.
I wonder if there is a concept for ...
2
votes
0answers
46 views
Isomorphism of $P(V)$ and $P(V^*)$
Let $V$ be a finite-dimensional left vector space over a division ring $K$, and let $V^*$ the dual right vector space (consisting of all linear functions from $V$ to $K$). We can (and will) treat ...
2
votes
0answers
42 views
Continuity of linear form
Let $E=\mathbb{R}[X]$
We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$.
Help me to study the ...
2
votes
0answers
43 views
Randomized Solution to a System of Inequalities
Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not:
$$
\mathbf x \le \sum_{i=1}^n \alpha_i ...
2
votes
0answers
68 views
Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?
A lot of known quantum gates are in the Pauli group $(I, X, Z, Y)$ or in the Clifford group $(H, P, Cnot)$. I need examples of the quantum gates that aren't in this groups. Also, are there are matlab ...
2
votes
0answers
83 views
Is the notion of a dual space related to the set of polynomial functions on an affine algebraic variety?
Let $M$ be an affine algebraic variety and consider the ring of polynomial functions on $M$, $\mathcal{O}(M):=\{f: M\to k : f\text{ a polynomial}\}$. If $k$ is algebraically closed we can recover our ...
2
votes
0answers
558 views
How can I find two vectors in a given span {u, v} that are not multiples of u or v?
But do appear to be linear combinations? $u$ and $v$ are 3-component vectors. The question posed is:
Find two vectors in span{u,v} that are not multiples of u or v and show the weights on u and v ...
2
votes
0answers
84 views
Condition on $c$ for a contraction map
I am extending Example 2.2 on this sheet.
Suppose $f(x(s),s)$ is such that $|f(x(s),s)-f(y(s),s)|\leq K |x-y|$ for some $K>0 ---(1)$
and $x,y\in C[0,t_f]:\,\,\,t_f<\infty$
Also, let $T$ ...
1
vote
0answers
13 views
Are there nonlinear operators that have the group property?
To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on ...
1
vote
0answers
52 views
Operations on vector spaces
Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
1
vote
0answers
25 views
Hahn Banach to get linear functional bounded by sub/superlinear functionals
I am working in a real vector space $V$. I have seen it written that if I have a sublinear functional $p$ and a superlinear functional $q$ such that $q \le p$ then there exists some linear functional ...
1
vote
0answers
30 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
1
vote
0answers
35 views
Proving that the circumcenter is the centroid
Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
1
vote
0answers
26 views
Difficulty understanding the definition of the Barycentric coordinate system
Specifically, the definition at http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29#Definition
Let $x_1, \dots , x_n$ be the vertices of a simplex in a vector space $A$. If, for ...
1
vote
0answers
44 views
Direct (Inductive) limit of (locally convex) TVEs and universal property
This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs.
I'm trying to work with universal properties in Topological ...
1
vote
0answers
34 views
To show that something is a four-vector
I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
1
vote
0answers
18 views
Is the polar decomposition useful in the real case as well?
I'm reading Roman's Advanced Linear Algebra p.252, where he talks about the Polar Decomposition. He states the theorem only for the case of $V$ a complex inner product space. Wikipedia also states the ...
1
vote
0answers
28 views
It is not possible to introduce multiplication in $v_n$(For $n>2$) so as to satisfy all field properties
In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells
It can be shown that except $n=1, 2$, it is not possible to introduce multiplication ...
1
vote
0answers
23 views
Reverse of rotate a vector around an axis - Finding North with magnometer
I've been reading the wikipedia page on rotation matrices and I know the reverse (or at least a very related) version of this question has been asked many times before on this site. However my ...
1
vote
0answers
22 views
A modular which is not a metrizing modular (hence not an F-norm)?
I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces.
Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
1
vote
0answers
48 views
Vector space-minimal polynomial
Let $V$ and $W$ be finite-dimensional vector spaces over $R$ and let $T_1 \colon V \rightarrow V$ and $T_2 \colon W \rightarrow W$ be linear transformations
whose minimal polynomials are given by ...
1
vote
0answers
21 views
vector analysis and reduce to a general expression in theta's
$\vec{A},\vec{B},\vec{C}$ and $\vec{D}$ are unit vectors ($|A|=1,|B|=1,|C|=1$ and $|D|=1$). The angle between the vectors,
1) $\vec{A}$ and $\vec{B}$ is $\theta_{1}$ ...
1
vote
0answers
31 views
Field structure of vectors in $\mathbb{R}^3$
Probably a trivial question: By representing vectors in $\mathbb{R}^2$ as complex numbers we can define multiplication of vectors so that $\mathbb{R}^2$ has a field structure. Can this be extended to ...
1
vote
0answers
84 views
Vector Projection and Cross Product
Let there be $w, u, $ and $v$, such that:
$$w \times u = \langle1, 3, 5\rangle$$
$$w \times v = \langle 2, 4, 6\rangle$$
Find:
$$v \cdot (((u \times w) \times v) + \text{VP}uv(w)) + ((u ...
1
vote
0answers
43 views
Strictly convex absolutly 1 homogeneous function
Is every strictly convex, 1-homogeneous function on $\mathbb R^d$ simply a multiple of the Euclidean norm?
Update: The above is no, since any p-norm on $\mathbb R^d$ is strictly convex and ...
1
vote
0answers
52 views
Vandermonde matrix
Let ${\bf G} \in\mathbb{C}^{M\times K}$ and ${\bf H} \in\mathbb{C}^{N\times K}$ are full-rank Vandermode matrices where $MN-1=K>N\geq N$, that is, ${\bf G}$ and ${\bf H}$ are fat. Let ${\bf F}= ...
1
vote
0answers
53 views
Changing variables
If there is a change of variables:
$$(\vec x(t),t)\to (\vec u=\vec x+\vec a(t),\,\,\,v=t+b)$$ where $b$ is a constant.
Suppose I wish to write the following expression in terms of a gradient in ...
1
vote
0answers
71 views
transformation of symplectic structure by a matrix
Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have
$$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$
Let ...
1
vote
0answers
40 views
Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?
Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces:
$$
H = H_1 \otimes \cdots \otimes H_n,
$$
and let $\mathcal{H}$ be a ...
1
vote
0answers
117 views
Gradient Descent for Primal Kernel SVM with Soft-Margin(Hinge) Loss
Given the primal objective
$$F({\bf a})=L\sum_{i,j}a_{i}a_{j}k(x_i,x_j) + \sum_{i}max(0, 1-y_i \sum_{j}a_jk(x_i,x_j)$$
for the soft margin SVM, where ${\bf a}=(a_1,...,a_N)$, N being the number of ...
1
vote
0answers
31 views
Coordinate Transform partials question
I wish to go from cartesian to cylindrical coordinates using the chain rule. I see here that
$x = rcos(\phi) $
$y = r sin(\phi)$
$r = \sqrt{x^2 + y^2}$
$ \phi = arctan(\frac{y}{x})$
I am ...
1
vote
0answers
69 views
Vector fields generating a transformation
It would be great if someone can explain to me what the following means:
Vector fields $V_i, i=1,2,3$ generate 3 single-parameter groups of transformations in $\mathbb R$ -- $$\tilde x ...
1
vote
0answers
27 views
Showing every finite dimensional subspace of a comodule lies in a finite dimensional subcomodule
Let $k$ be a field and let $(C, \Delta, \epsilon)$ be a vector space which is a coalgebra. Let $(M, \delta)$ be a comodule. Suppose $V \subseteq M$ is a finite dimensional space. For each $v \in V$, ...
