# Tagged Questions

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### Perturbation theory for algebraic equations

I'm trying to find expansion (up to the 2nd non zero term) for the roots of: $x^5-x^2+\epsilon=0$ as $\epsilon\rightarrow0$ So I've assumed the solution may be written as a power series ...
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### Polynomial vector space terminology

Consider the vector space $P$ and the subset $V$ of $P$ consisting of those vectors (polynomials) $x$ for which a) $2x(0) = x(1)$, b) $x(t) = x (1-t)$ for all $t$. In which of these cases is $V$ a ...
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### What am I doing wrong? Gram Schmidt process..

Let there be the inner product of all polynomials of degree smaller or equal to 2: $\langle f,g\rangle=\int_0^1f(x)g(x)xdx$. Find orthonormal basis. So I really tried this for an hour and it pretty ...
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### Show that a set of polynomials make a linear space.

I have a problem that states: "Let P be the set of all polynomials of degree at most 2. Show that P is a linear space." I know how to show that a set of vectors make a linear space with a certain ...
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### Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
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### How to find charpoly from eigenvalues and CH to prove an equation

For an uknown 3x3 matrix $A$ we know that $\operatorname{tr} A = 0$, $\det(A) = 1/4$ and we also know that two eigenvalues are the same. Proove that $4A^3 = -3A - I$. Problem says to use Vieta to find ...
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### sufficient and essential condition for $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ [closed]

What is the sufficient and essential condition for two real polynomials $P(x)$ and $Q(x)$, such that $P(\sin x)= Q(\cos x)$ for $x\in (\alpha, \beta)$, $\alpha\lt \beta$?
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### A question on Arithmetic Progressions is given in the picture below..

What is the sum of an arithmetic progression whose first term is $a$, the second term is $b$, and the last term is $c$? A. $\dfrac{(b+c-2a)(a+c)}{2(b-a)}$ B. $\dfrac{(b+c+a)(a+c)}{2b-a}$ C. ...
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### Matrix for linear map involving polynomials

I need to find the matrix corresponding to the linear map $f:V_3 \rightarrow V_3$, where $V_3$ is the vector space of all polynomials of degree less than or equal to 3, $$f(p(X))=p(X)-p'(X)$$, with ...
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### A question on quadratic equations.. Given below in the picture.

PLease also tell how u got to the answer as I want to know the way to solve further questions
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### find vector x so that vector: diag(x).C.x has all components equal, where C is positive-definite

PROBLEM: I am trying to find closed form solutions or provable general properties of solutions for the solution $x$ of the following: Find $\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{bmatrix}$ ...
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### Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
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### How to convert a permutation to permutation polynomial?

Let Fq be the finite field with q elements, where q is a prime power. A permutation on Fq is a bijection from Fq to itself. Let Fq[x] be the ring of polynomials in a single indeterminate x over Fq. A ...
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### Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
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### If $(A-\lambda{I})$ is $\lambda$-similar to $(B-\lambda{I})$ then $A$ is similar to $B$

When reading the topic about primary and rational canonical form of matrices I stuck myself on this theorem: The matrices $A,B\in K^{n\times n}$ are similar if and only if their characteristic ...
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### Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
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### Nondiagonalizable Matrix and Polynomials

I got the following problem: If $A$ is a nondiagonalizable square matrix of order $n$ over field $\mathbb{F}$ then there exists a polynomial $P$ of degree $n-1$ over $\mathbb{F}$ such that ...
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### Minimal Polynomial of a scalar multiple of a Matrix

I got the following problem: Let $A$ be a square matrix of order $n$ over field $\mathbb{F}$ and let $M_A$ be the minimal polynomial of $A$ of degree $k\in\mathbb{N}$ and let $0\neq c\in\mathbb{F}$ ...
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### polynomial fractions $\left(\frac{1}{n+x}\right)_{n \in \mathbb N}$ with domain $\mathbb R_{>0}$ linear independent?

My approach: let $f_n(x):=\dfrac{1}{x+n}$. Then we want to check for which $\lambda_i$'s the following holds. $\sum \limits_{i=0}^{n} \lambda_i f_i=0$. By multiplying the denominator out we get the ...
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For a $A\in M_n({\bf C})$ with a minimal polynomial $m(x) = (x-c)^n$ then we have a Jordan form wrt some basis  A=\left( \begin{array}{ccccc} ...