1
vote
1answer
52 views

Understanding the term “Abstraction” in mathematics

When the need for abstraction is asserted in mathematics is it generally meant that there is a need to apply a definition to n-dimensions such that n is an integer going to infinity?
0
votes
1answer
32 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
2
votes
2answers
58 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
0
votes
1answer
56 views

Particular solution of a system of linear differential equations

Let $A(t) \in \mathbb R^{2\times 2}$ and $b(t) \in \mathbb R^2$ continuous functions in an open interval $I$. Consider the system $$(1) \space X'=A(t)X+b(t).$$ Let $X_1,X_2$ be linearly independent ...
1
vote
2answers
29 views

Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
2
votes
1answer
30 views

Surjective function from a set of funtions to itself

Let a function be defined by $f \longmapsto f'$ acting from the set of all polynomials to itself. I am asked if this is surjective. I would like to think it isn't, but I'm in doubt how I should ...
0
votes
1answer
27 views

What is the necessary and sufficient condition of linear dependence of $n$ functions?

If $n$ functions are linear dependent, then the Wronskian determinent is zero, While that the Wronskian determinent is zero cannot imply $n$ functions are linear dependent. So what is the necessary ...
1
vote
1answer
39 views

Scalar-by-matrix Derivative of Quadratic Product

I'd like to know $\frac{\partial f(\mathbf{U})}{\partial \mathbf{U}}$, i.e., the 'by-matrix derivative' of the following scalar function $f(\mathbf{U})$ w.r.t. $\mathbf{U}$. $$f(\mathbf{U}) = ...
7
votes
1answer
188 views

Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
1
vote
0answers
11 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
0
votes
1answer
23 views

Question related to vector space of solutions of a differential equations system.

I have some doubts regarding the proof of the statement: The set of solutions of the system $$X'=A(t)X$$ where $A(t):I \subset \mathbb R \to \mathbb R^{n \times n}$ is continuous, forms an ...
1
vote
0answers
29 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
0
votes
1answer
30 views

Alternative definition of Euclidean operator norm

Given $A\in \mathbb{C}^{n\times n}$, $\|.\|$ the Euclidean operator norm, and $\rho(A)$ the spectral radius of A, how to show that $$ \|A\| = \sup\{\rho(AB):B\in \mathbb{C}^{n\times n}, \|B\|=1\} $$
0
votes
1answer
26 views

‎every ‎ring ‎automorphism‎ $\Phi$ ‎of ‎the ‎complex ‎algebra ‎‎

Why ‎every ‎ring ‎automorphism‎ $‎\Phi$ ‎of ‎the ‎complex ‎algebra $‎‎M‎‎‎_{‎n‎} ‎(‎\mathbb{C}‎)$ ‎of ‎all $‎n‎\times n$‎ ‎complex ‎matrices ‎has ‎the ‎form ‎‎ ‎$\Phi ‎(T)= ‎AT‎A‎^{-1} ‎‎$ ?
0
votes
1answer
31 views

Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
1
vote
0answers
25 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
2
votes
2answers
44 views

All unit vector has bounded components?

If $\Vert v_i\Vert \leq 1$ for all $1\leq i\leq k$ with $\{ v_1,..,v_k\}$ is linearly independent THEN FOR ALL real numbers $\alpha_i$ with $$\Vert \sum_{i=1}^k\alpha_iv_i\Vert=1$$ we can find ...
0
votes
1answer
53 views

Jacobian matrix and determinant - relation to orientation

$F$ is a function from $V$ to $V$ where $V$ is a $n$-dimensional vetor space and $p \in V$. In the article Jacobian determinant it says: "If the Jacobian determinant at $p$ is positive, then $F$ ...
2
votes
3answers
160 views

When to use $\times$ and $\otimes$

Im wondering when to use $\times$ and when to use $\otimes$. In some cases it seems very straightforward, for example $\times$ can be used when combining two elements into an n-tupel (for a product ...
0
votes
0answers
32 views

Eigenvalues of the linear operator $T^*T$

Let $T$ be a linear operator $T: H_1 \mapsto H_2$, where $H_1$ and $H_2$ are both Hilbert Spaces. Suppose further that $T$ is bounded, but not self adjoint. Suppose I also know that for functions ...
2
votes
1answer
266 views

How to make this polynomial the zero polynomial?(recursively)?

Given a fixed $\beta \in \mathbb{R}$, I want to find the $c_0,...,c_n$ for arbitrary $n \in \mathbb{N}$ such that the polynomial \begin{align}P_n(z):=z(1-z) ...
0
votes
0answers
15 views

Generalized Binomial Equality

I want to show that, assuming $p>0$, $(2^p+1)^{2/p}=1+2^{1+2/p}$ iff $p=2$. I want to do this in order to show that the norm on $\mathbb R^2$ given by $||(x, y)||=(x^p+y^p)^{1/p}$ arises from an ...
1
vote
3answers
131 views

Choosing good textbooks in linear algebra, analysis and graph theory

I need some advices to choose good undergraduate textbooks in LINEAR ALGEBRA, ANALYSIS and GRAPH THEORY. I found: Gilbert Strang // Introduction to Linear Algebra - Welleslay Cambridge Press (2009) ...
4
votes
4answers
169 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
1
vote
1answer
22 views

Notions of tangent plane at function

For a differentiable function $f : \mathbb R \to \mathbb R$ the equation of the tangent plane at $x_0$ is $0 = f'(x_0) x - y$. But some functions not differentiable like $\sqrt x$ at $x_0 = 0$ still ...
1
vote
0answers
23 views

Exercise regarding normal matrices and their spectrum

I hope could get a few hints to this exercise Let $T\in M_n (\mathbb{C})$ be a normal matrix. Let $\lambda \in \sigma(T)$, where $\sigma(T)$ is the spectrum of $T$. Argue that $1_{\{\lambda\}}\in ...
0
votes
0answers
29 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
0
votes
2answers
40 views

Show two vectors are linearly independent

So I need help with this problem! I am confused because there is only one equation? I tried writing it in form $af(x) + bg(x) = 0$ but I really am quite stuck. Any help is greatly appreciate.
2
votes
1answer
93 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
0
votes
3answers
56 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
1
vote
1answer
192 views

Is the inverse of a symmetric positive semidefinite matrix also a symmetric positive semidefinite matrix?

If we let $$S_{++}^n(\mathbb{R})$$ denote the set of all square symmetric positive definite matrix over the real numbers, then is it true if $A\in S_{++}(\mathbb{R}) \implies A^{-1} \in ...
4
votes
1answer
75 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
0
votes
0answers
18 views

any theory about determinant as a function of the elements of a matrix

Let $X=(x_{ij})_{1\le i,j\le n}$ be a n by n matrix where $n\ge3$. Consider the function $f:(R^n)^n\to R$ given by the formula $f(X):=\det(X)$. (a) Assume that $rank(X)=n-1$. Is it true that X is a ...
1
vote
2answers
39 views

Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
0
votes
1answer
32 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
1
vote
1answer
42 views

To prove that these matrices are invertible

Let $A$ and $B$ be $n \times n$ matrices such that $||I - AB|| < 1$. Prove that $A$ and $B$ are invertible, and $$A^{-1} = B \sum\limits_{k=0}^{\infty} (I - AB)^k \text{ and } B^{-1} = ...
1
vote
2answers
44 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
2
votes
1answer
28 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
0
votes
1answer
31 views

Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent

Let P be a vector space of polynomials with real coefficients. Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent, where $|p|_1$=max$ \{|p(t)|$; $0\leq t \leq 1 \}$ and $|p|_2$ = max ...
3
votes
1answer
55 views

difference between weak* convergence and convergence

I am trying to prove the following: If $X$ is a finite-dimensional space, then for sequences $\left\{x_n\right\}\subseteq X$ and $\left\{f_n^*\right\}\subseteq X^*$, if there exists an $x\in X^*$ ...
0
votes
1answer
29 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
3
votes
0answers
49 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
3
votes
2answers
140 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
1
vote
2answers
54 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
1
vote
1answer
24 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
1
vote
1answer
38 views

Understanding a Proof for Why $\ell^2$ is Complete

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I'm trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I'm trying to show that $\ell^2$ is ...
3
votes
1answer
116 views

How to solve this system of 3 equations with 3 variables?

I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$: \begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ ...
2
votes
0answers
56 views

Exterior power of a space of maps $(\mathbb{K}^T)$

We are given a set $T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K}$ Could you help me prove that if $ \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in ...
1
vote
2answers
49 views

Eigenvalues and eigenvectors of an operator

I have $Ku(t)=\int_0^1 G(t,s) u(s) ds, u\in L^2[0,1]$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ ...
2
votes
2answers
31 views

clarification on a question about showing that the closure of a subspace is a subspace

In a homework problem, I have been asked to prove the following "If $X$ is a normed linear space and $S$ is a linear subspace of $X$ then $\overline{S}$ is a linear subspace of $X$." ($\overline{S}$ ...