# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$$ ...
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### 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 $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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### 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!
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, $P_a(t) = t^2 -2t +1 -a$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T$?
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### 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 $\langle f,g\rangle := \int_{-1}^1 f(t)\overline{g(t)}dt$. ...
### 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 ...
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) ...