1
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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 ...
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) ...
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 $P(t) \in V \Leftrightarrow (t − a)^2$ ...
0
votes
2answers
23 views

About the subspace of polynomial vector space

Why the set of functions in $C\left [ 1,-1 \right ]$ such that $f\left ( -1 \right )= f\left ( 1 \right )$ is the subspace of $C\left [ 1,-1 \right ]$?
1
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0answers
25 views

Is there a computationally efficient way to find the part of a vector, which is of certain order in independent variable x?

Let $\vec{a}$ be an element of a vector space over the space of monomials, i.e. $$ \vec{a}\left(x\right)=\sum_{j=1}^{N}a_jx^{k_{j}}\vec{e_{j}} $$ Remark: For simplicity, here we operate with only ...
0
votes
4answers
87 views

Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
0
votes
1answer
35 views

Let $K = $ algebraic numbers. Then is $\operatorname{Span}_K(\pi, \pi^2, \dots)$ a vector space of transcendentals?

$V = {\rm Span}_K(\pi, \pi^2, \dots)$ is clearly a $K$-vector space. If we let $K = \Bbb{Q}$ temporarily, then every element of $V$ is transcendental as it's a finite linear combination $Q(X), \ X = ...
3
votes
3answers
82 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 ...
1
vote
2answers
216 views

Determine whether S is a subspace of P3. Vector space of all real polynomials.

ATTEMPT: Have given a small attempt just really confused on how to approach. So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$. So we find the derivative? and find the values of ...
0
votes
1answer
52 views

Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
1
vote
0answers
44 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
1
vote
1answer
28 views

What is fixed in a equation in a polynomial vector space

From what I've learned, an equation $p(t)$ in $P_n$ is defined $$p(t) = a_0+a_1t+a_2t^2+\cdots+a_nt^n \tag 1$$ Given the basis $\beta=\{1,t,t^2,\ldots,t^n\}$, $p(t)$ can be written in the form $$p(t) ...
2
votes
1answer
187 views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
1
vote
2answers
35 views

Show inequality for two elements in $\mathbb{R}^n$

I know that $x,y\in \mathbb{R}^n$ are such that $x_1\leq0,x_1^2\geq x_2^2+\dots+x_n^2$ and $y_1\geq 0,y_1^2\geq y_2^2+\dots+y_n^2$. Is it possible to show that $$x_1y_1+x_2y_2+\dots +x_ny_n\leq 0$$ ...
1
vote
2answers
49 views

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 ...
0
votes
1answer
46 views

Distance from Vector to the Linear Span

Let $V$ be the space of real polynomials of degree $\leq n$. a) Check the setting $(f(x),\,g(x))=\int_{0}^{1}f(x)g(x)\,dx$ turns $V$ to a Euclidean space. b) If $n=1$, find the distance from ...
2
votes
1answer
26 views

Clarification regarding notation: a $\mathbb{C}[X]$ -basis of $\operatorname{Der}_{\mathbb{C}}\mathbb{C}[X]$

I came across the following sentence in an article: The derivations $\frac{\partial}{\partial F_1},\ldots,\frac{\partial}{\partial F_n}$ form a $\mathbb{C}[X]$-basis of ...
2
votes
0answers
88 views

Find all the invariant subspaces of T

T is a linear transformation, defined as the following: $T(p(x)) = xp(x)$, $T\colon R[X]\to R[X]$ Find all the invariant subspaces of $T$. As I see it, only the trivial subspaces $0$, $R[X]$ are ...
1
vote
0answers
101 views

Determining whether the quotient space of the polynomials is finite.

I'm just a little unsure whether my answers to this question are right. Let $V=F[x]$ be the vector space of polynomials over the field $F$. Determine whether or not $V/M$ is finite dimensional when ...
0
votes
0answers
39 views

Picking out Vector Coefficients in Magma

Suppose $K/k$ is a finite field extension, with fixed basis $e_1,\ldots,e_d$, and $f\in K[x_1,\ldots,x_n]$. Using our basis, $f$ has a unique expression as a sum of polynomials in ...
1
vote
1answer
85 views

Prove that the determinant of polynomials is zero

Prove that this determinant is zero (this matrix is $n\times n$): $$\begin{vmatrix} f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\ f_2(a_1) & f_2(a_2) & \cdots & f_2(a_n) \\ \vdots ...
1
vote
0answers
122 views

The vector space of polynomials

I was given a theorem: The polynomials (where $f$ and $g$ are complex polynomials of degrees $n$ and $m$) $$f(z), zf(z), \ldots , z^{m−1}f(z), g(z), zg(z), \ldots,z^{n−1}g(z)\tag{7.6.4}$$ ...
3
votes
3answers
88 views

dimension of space of polynomials

Let $\mathcal P_k^n$ be the space of all polynomials of degree $\leq k$ in $n$ variables. Prove $\dim\mathcal P_k^n = {n+k\choose k}$. I tried showing this by taking $n\in\mathbb N$ an arbitrary ...
2
votes
1answer
68 views

Find Eigenvectors of a homomorphism over polynomial vector spaces

Let $\mathbb{R}[x]$ be the real-valued vector space of polynomials with real-valued coefficients and $F: \mathbb{R}[x]\rightarrow\mathbb{R}[x]$ be a homomorphism defined as $$ ...
1
vote
0answers
51 views

A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$

Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$. $F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$. $P_1(F)$ is the set of all ...
0
votes
1answer
37 views

Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$

I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
0
votes
1answer
50 views

$T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$

So, I'm asked to give an example of a linear map $T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$. As far as I understand, I'm trying to ...
1
vote
1answer
66 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
2
votes
6answers
288 views

Proof that $\mathbb{R}[x]$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?
4
votes
1answer
1k views

Dimension of Vector Space (Polynomial)

I was asked by a friend to: "Find the dimension of the vector space consisting of all polynomials in $n$-variables of degree at most $k$".Now, my response to him was that since the basis consists of ...
1
vote
1answer
1k views

Linear Algebra span of vectors with polynomials and linear independence

So the one below is not a matrix they are vectors. Im still not used to writing vectors. So Im trying to to find the values of k for which the vectors are linearly independent. Well what I know is Ax ...
5
votes
4answers
450 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
1
vote
2answers
160 views

Vector space of polynomials with given root

Let $\mathbb{R}[t]_{\leq n}$ represent the vector space of polynomials (with coefficients in $\mathbb{R}$) whose degree is at most $n$. $\forall a \in \mathbb{R}$, let $U_a = \{P(t) \in \mathbb{R}[t] ...
2
votes
2answers
80 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
0
votes
1answer
116 views

Show that (vector) subspaces of $\mathbb{A}^n$ are algebraic sets

i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of ...
2
votes
3answers
147 views

Explanation to the details of the proof that $F[x]$ is not finite-dimensional.

I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor. The space of polynomials $F[x]$ is ...
3
votes
2answers
744 views

Scalar Product for Vector Space of Monomial Symmetric Functions

Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $$ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $$ Since ...
0
votes
1answer
85 views

Linear Algebra: Minimum Polynomials for Operators on $\mathbb{R}[x,y]$.

I've been working through some problems set by my University over the past few years, and have encountered this problem. Problem Let $n > 1$ and let $V_n$ be the subspace of $\mathbb{R}[x, y]$ ...
1
vote
1answer
643 views

Linear Algebra: Dual Basis Problem

Problem Let $V$ be the vector space of all polynomial functions $p$ from $\mathbb{R}$ to $\mathbb{R}$ which have degree two or less. Define three linear functionals on $V$ by ...
0
votes
1answer
670 views

What is meant by "All Polynomials of the form $p(t) = a + t^2$?

I have a math homework problem that goes like this: Determine if the given set is a subspace of $\mathbb{P}_n$ for the appropriate value of $n$: All polynomials of the form $p(t) = a + t^2$, where ...
2
votes
1answer
6k views

Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n?

Definition. A subspace of a vector space V is a subset H of V that has three properties (a) The zero vector of V is in H. (b) $\forall$ u, v $\in H$, we have u + v $\in H$. (c) $\forall$ u $\in ...