0
votes
1answer
16 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
0
votes
1answer
13 views

dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
1
vote
2answers
31 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
12 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
33 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
26 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
vote
0answers
29 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
93 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
91 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
377 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
48 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
31 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
253 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
53 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
56 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
98 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
117 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 ...
1
vote
1answer
91 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
136 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
96 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
74 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
51 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
69 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
347 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
2k 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
473 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
164 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
90 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
123 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
150 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
769 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
86 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
686 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
683 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 ...