3
votes
3answers
101 views

Basis in the vector space of all polynomials

Let $V$ vector space of all polynomials $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a ...
0
votes
3answers
84 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
1
vote
1answer
44 views

About $\mathbb{F}_7[x]$

can you help me with this? Let $a(x)=3x^6+2x^2+x+5$ and $b(x)=6x^4+x^3+2x+4$, find the g.c.d between $a(x)$ and $b(x)$ in $\mathbb{F}_7[x]$. Thanks!
0
votes
2answers
60 views

Matrix with rank 3 does not exist in this $p(x)$

Given: Characteristic polynomial is $p(x) = x^7 - x^5 + x^3$ . Prove that there isn't a matrix A that $ \rho(A) = 3 $ I tried to play with $p(x) = x^3(x^4 - x^2 +1)$ But I'm still not sure how ...
1
vote
1answer
45 views

Characteristic polynomial $p_{cA}(t)$

Let's define $p_{A}(t)$ the characteristic polynomial of square matrix $A$ over $R$. Prove that for every $c \in R$, $c \ne 0$ the characteristic polynomial $p_{A}(t)$ of the matrix $cA$ is ...
0
votes
4answers
69 views

Prove that the characteristic polynomial of a nilpotent matrix is $x^n$

How can I prove that the char.pol. of a nilpotent matrix is of the form $x^k$? I'm trying to do it by contradiction but assuming that $p_{xA}=a_0+a_1x+\dots+a_mx^m+\dots+a_nx^n$ seems not giving any ...
1
vote
4answers
42 views

find f(x) polynomial with rational coefficients such that $f(x)^{2} = g(x)^{2}(x^{2}+1)$

g(x) is a polynomial with rational coefficients that is not 0 . I need to find f(x) polynomial with rational coefficients such that: $f(x)^{2} = g(x)^{2}(x^{2}+1)$ or prove such polynomial does not ...
0
votes
1answer
38 views

prove that if T is invertible transformation there is polynomial $p$ such that $T^{-1} = p(T) $

I know how to prove this using Hamilton.C but something doesn't make sense to me. if I assume that there is such polynomial p(x), so p(T)T = I . then looking at these polynomials I get: p(x)x = 1 so ...
0
votes
1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
0
votes
1answer
33 views

Polynomial function with 2 points

Can someone please help me solve this problem ? ...
0
votes
3answers
40 views

Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix} $ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a $ Then, $ P_a(t) = t^2 -2t +1 -a$ ...
0
votes
0answers
18 views

Find canonical form of bi-linear form on polynomials

$V$ is vector space of polynomials in degree less or equal than $2$, we define the bi-linear form: $f(p,q) = p'(-1)q(2$) where $p$ and q are polynomials from $V$. I need to find the canonical form ...
1
vote
2answers
42 views

Help understanding the characteristic polynomial

Let $$A = \begin{bmatrix} 1 &2 &1 \\ 2 & 2 &3 \\ 1 & 1 &1 \end{bmatrix}$$ I'm calculating the characteristic polynomial by the following: $$P(x) = -x^3 + Tr(A)x^2 + ...
0
votes
2answers
64 views

Calculating the characteristic polynomial

I'm stuck with this problem, so I've got the following matrix: $$A = \begin{bmatrix} 4& 6 & 10\\ 3& 10 & 13\\ -2&-6 &-8 \end{bmatrix}$$ Which gives me the following ...
2
votes
2answers
22 views

Calculate the angles between $(1,X),(X,X^2),(X^2,X^3),(X^3,X^4)$ given the inner product $\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$

Let $V_4$ be the vector space of all polynomials of degree less than or equal to 4 with the inner product $$\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$$ calculate the angles between ...
2
votes
1answer
80 views

Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T

$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\LM}{\mathcal{L}}$ Question: Suppose $\F = \C, T \in \LM(V), p \in ...
3
votes
2answers
75 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
-3
votes
1answer
41 views

Finding an eigenbasis of an operator.

Consider the operator $T:\mathbf{P}_1\rightarrow\mathbf{P}_1$ by $T(ax+b):=-bx+(2a-3b)$. I want to find out if there is a basis of eigenvectors of this operator $T$. Now, I have no idea what to do, ...
3
votes
1answer
36 views

Angle between two polynomials

Given the inner product of two polynomials $p(X), q(X) \in P(d)$, where $P(d)$ is the vector space of all polynomials of degree less than or equal to d, with real coefficients, and using the inner ...
4
votes
4answers
183 views

Prove that p has m distinct roots if and only if p and p' have no roots in common

Problem: Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p'$ have no roots in common. My proof so far: If $m=0$, ...
1
vote
0answers
21 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
1
vote
1answer
64 views

Exercise about basis

I am trying to solve the following exercise: Let $A \in L(P_3)$ be defined by a matrix: where $A^b_e$ = $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ ...
0
votes
1answer
34 views

Linear independence of a set of 'simple' functions

I have some $\alpha = \{f_0(x),f_1(x),\dots,f_n(x)\}$ Where having the $k$ in $f_k(x)$ equal $x$, meaning $k=x$ makes $f_k(x)=1$ or zero if $k \ne x$. Now I want to show $\alpha$ is linearly ...
0
votes
1answer
27 views

Set of polynomials

I want clarification on the following question: Let $\{c_0,c_1,c_2,\dots,c_n\}$ denote a set of $n+1$ distinct elements in $\mathbb{R}$. Define the set of $n+1$ polynomials. $$f_j(x)=\prod_{k=0,k\ne ...
1
vote
1answer
73 views

Is my proof correct? linear transformation over $\Bbb R$ has invariant subspace $\dim(U)=1$ or $2$

$V$ is a Vector space over $\Bbb R$, and $\dim(V)=n$. A linear transformation $T$ from $V$ to $V$. Then, T has an invariant subspace $U$ such that $\dim(U)=1$ or $2$ I read in many books, which ...
0
votes
0answers
30 views

How to determine if a basis is top-down?

I am unclear as to what the rules are that define a set to be a top-down basis. For example: what are top-down basis for P2? Below is the grid of polynomials for P2. \begin{array}{ccc} (t-1)^2 ...
0
votes
1answer
22 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
1
vote
4answers
19 views

What is the matrix corresponding it a linear transformation of a polynomial?

Given the linear map $T(f(x)) = f(2x+1)$ where $f(x)$ is a polynomial of degree $3$, what is the matrix corresponding to $T$?
1
vote
1answer
13 views

Find basis for $P_0^\perp \subset P_4$ and $\ker (f \mapsto f(0))$ in $P_4$

Let $P_n \subset \textrm{Map}(\mathbb{C},\mathbb{C})$ the space of polynomial maps $\mathbb{C} \to \mathbb{C}$ with degree $\le n$. We define $\langle f,g\rangle := \int_{-1}^1 f(t)\overline{g(t)}dt$. ...
1
vote
3answers
22 views

Let $f: V_3 \rightarrow V_3$ be the function such that $p(X) \mapsto p''(X)$, calculate the eigenvalues of f

Let $V_3$ be the vector space of all polynomials of degree less than or equal to 3. The linear map $f: V_3 \rightarrow V_3$ is given by $p(X) \mapsto p''(X)$. Calculate the eigenvalues of f. First of ...
2
votes
1answer
252 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) ...
2
votes
3answers
30 views

Finding the characteristic polynomial in a square matrix

The example in the textbook had a square matrix \begin{pmatrix} 0&1&0\\0&0&1\\4&-17&8 \end{pmatrix} Then proceed to say $ \ (\lambda \cdot I - A) \ $ is \begin{pmatrix} ...
1
vote
0answers
20 views

Dual Vector Spaces - Evaluation at a Point of a Polynomial Gives a Basis

My question is the following: Let $\{a_0,a_1,...,a_n\}$ be (pairwise) distinct, real numbers. Let $V$ be the vector spaces of all polynomials of degree at most $n$, ie $V = \Bbb P_n$. Let $\phi_j : ...
0
votes
1answer
53 views

Finding polynomial from sum and product of zeroes

The question is Form a quadratic polynomial whose sum of zeroes are $-1/3$ and product of zeroes are, $4$. I did this, $\alpha + \beta = \frac{-b}{a} = \frac{-1}{3}$ $\alpha \cdot \beta = ...
2
votes
2answers
72 views

Zeroes of polynomial

$$c_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$ Prove that for every polynomial with integer ...
5
votes
1answer
85 views

Solving Cubic Equations Using Origami

I have to write a research paper on a mathematical topic for my class; I chose the above topic. I understand that a parabola can be formed using a focus and directrix, both created by origami folds, ...
0
votes
1answer
19 views

relationship between number of polynomials and dimension of the space.

If p1,p2,...,pk are linearly independent polynomials in Pn, a mathematical relationship between k and n is: k<=n. If the k will be more than n, the set of polynomials can not be linearly ...
-1
votes
1answer
16 views

Show that $ M_f$ is a factor of $ C_f$

Let $V$ be a finite dimensional vector space over a field $K$. Let $f:V \rightarrow V$ be a linear map. Let $M_f$ be the minimum polynomial of $f$ and let $C_f$ be the characteristic polynomial of ...
1
vote
1answer
29 views

Dimension of subspace in the space of polynomials

Let $W$ be the linear space of polynomials in two variables $x$, $y$ of degree at most $N$, i. e. $$ W=\{f\in\mathbb{R}[x,y]:\deg(f)\leq N\} $$ Consider subspace $$ V=\left\{f\in W: ...
0
votes
1answer
19 views

Representing a polynom with a base

How to do I present $$P(x)= 6x^2+4x+2$$ with the base $$B=\frac{1}{\sqrt2},\qquad x\sqrt\frac32,\qquad \sqrt\frac58(1-3x^2)$$
0
votes
1answer
79 views

Polynomial: Finding its value

If $a-b=3$, $a+b+x=2$, then find the value of $(a-b)\left(x^3-2ax^2+a^2x-(a+b)b^2\right)$ I could only substitute the value of $a-b$ there. I seriously want to try as much as I can on my own but ...
1
vote
0answers
25 views

Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
2
votes
0answers
38 views

Can I modify a polynomial to return only multiples of a given number?

I'm attempting to create a polynomial equation for a project of mine, with a shape similar to the following: $${3x^5\over500}+{x^4\over25}+x^3+40 x^2+100 x$$ However, one of my goals is to have the ...
0
votes
2answers
80 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
1
vote
2answers
49 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
2
votes
0answers
32 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
0
votes
1answer
82 views

dimension of subspace - polynominals evaluated on f

I need to prove that the dimension of the subspace of endomorphisms is less or equal m, if m is the degree of a polynomial p of K[t] \ {0} with p(f) = 0 (f is endomorphism). In a second step I ...
3
votes
3answers
78 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
2
votes
1answer
71 views

Inverse of a matrix is expressable as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
0
votes
2answers
32 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.