3
votes
0answers
20 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
0
votes
2answers
57 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
2
votes
1answer
45 views

Calculating Vandermonde determinant

I understand that the Vandermonde determinant $$ W(x_1, \ldots, x_n) = \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & ...
6
votes
4answers
377 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
2
votes
1answer
66 views

Method to simplify this long expression

How can I simplify this long expression: $-a^3(d-b)(d-c)(c-b)+b^3(d-a)(d-c)(c-a)-c^3(d-a)(d-b)(b-a)+d^3(c-a)(b-a)(c-b)$ I know that it is equal to $(d-a)(d-b)(d-c)(c-a)(c-b)(b-a)$ but i have no idea ...
1
vote
2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
1
vote
1answer
34 views

A question on the standard basis for polynomials

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book hasn't explained everything properly so my question might be extremely easy, apologize in advance: For ...
0
votes
0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
1
vote
3answers
71 views

Does substitute $\lambda$ with matrix $A$ in a polynomial conflict with the Axiom of Substitution?

This seems to be an elementary question, gonna ask it anyway. Suppose that $A$ is a square matrix, and that $p(x)$ is its characteristic polynomial, we know that (1) $p(x) = \det(xE - A)$ We also ...
0
votes
0answers
52 views

Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
0
votes
2answers
25 views

Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
0
votes
3answers
60 views

How to find a basis for a linear space of polynomials?

Questions such like, (1) Let $V = P^4$ be the vector space of all real valued polynomials of degree less than or equal to four. Let $W =\{p(x)\in P^3 |p(−2)=p(2)\}$. Find the basis for $W$ (2) Let ...
0
votes
1answer
35 views

Change of basis from falling powers to powers for polynomials up to degree $n$

Notice that $$(1, x, x^{\underline{2}}, x^{\underline{3}}, \dots)$$ and $$(1, x, x^2, x^3, \dots) $$ both are bases of $\mathbb{R}[x]$ (where $x^{\underline{n}}$ is the falling power). Now suppose the ...
5
votes
2answers
76 views

How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$?

How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$? Where $\alpha_1, \alpha_2,\ldots, \alpha_k$ are $k$ distinct ...
1
vote
3answers
61 views

Whether $\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$ is true

Suppose we have k positive numbers: $\alpha_1, \alpha_2, ..., \alpha_k$, for any number $\beta>0$, is $$\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$$ ...
0
votes
0answers
24 views

Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
1
vote
1answer
38 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
0
votes
2answers
47 views

Proof of subspace, finding a basis in polynomials

Let W = {(f(x)∈ P2[R] : f '(x) + xf(0) = 0} i) Prove that W is a subspace of P2[R]. ii) Find a basis for W. Here's what I have so far: i) I have to verify that ...
0
votes
4answers
68 views

Find the basis for the subspace of the set of polynomials of degree less than five?

Let U = {p $\in P_4(F): p(2) = p(5) = p(6)$. Find a basis for U. I know how to do this problem if I were given p(2) = p(5). Set the two equal to each other and solve for one of the coefficients. I ...
0
votes
1answer
19 views

How to write a polynomial basis with conditions

I don't understand how to do problem where you have to write a basis for a polynomial. For a example a typical problem would be something like: Let U = {p $\in$ $P_n(F)$: p(2) = p(5) or p''(1) = ...
3
votes
3answers
146 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
87 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
45 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!
-1
votes
2answers
61 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
47 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
80 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
44 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
42 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
34 views

Polynomial function with 2 points

Can someone please help me solve this problem ? ...
0
votes
3answers
41 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
46 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
70 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
24 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
83 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
91 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
52 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
38 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
201 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
76 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
1answer
24 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 ...
0
votes
4answers
20 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
265 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) ...