Vector spaces Definition and 286 Threads

It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC). Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory...

I'd like to discuss some aspects of quantum systems' state space from a mathematical perspective. Take for a instance a qubit, e.g. a two-state quantum system and consider the set of its pure states. This set as such is a "concrete" set, namely the "bag" containing all the qubit's pure states...

Does anyone have a reference that explains how the general linear group GL(n) acts on vector spaces and dual spaces? Furthermore, I would like to understand why the canonical pairing ##\langle\cdot, \cdot\rangle: V \times V^* \to \mathbb{F}##, ##(v,\alpha) \mapsto \langle\alpha,v \rangle :=...

picture since the text is a little hard to read i have no problem showing this is a vector space, but what is meant by complex dimention? Is it just the number on independant complex numbers, so n?

Hello! Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading As vector spaces they are obviously isomorphic since ##\dim_{\mathbb{R}}...

There is a passage in this book where I don't follow the logic; In this short quotation from 'Quantum Mechanics: The Theoretical Minimum' by Leonard Susskind and Art Friedman \mathcal{A} represents the apparatus that is performing the measurement the apparatus can be oriented (in principle) in...

##f : [0,2] \to R##. ##f## is continuous and is defined as follows: $$ f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$ $$ f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$ ##V = \text{space of all such f}## What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...

According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...

In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...

Hi PF, I've one question about vector spaces. There is only one way to define the operations of a vector space? For example if V is a vector space there is other way to define their operations like scalar multiplication or the sums of their elements and that the result is also a vector space?

Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an...

Solution 1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##. From...

This was a problem that came up in my linear algebra course so I assume the operation L is linear. Or maybe that could be derived from given information. I don't know how though. I don't quite understand how L could be represented by anything except a scalar multiplication if L...

I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about: For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?: d1 = Diagonal one = (a,b,c) d2 =...

[Moderator's Note: Spun off from previous thread due to increase in discussion level to "A" and going well beyond the original thread's topic.] A vector space has no origin to begin with ;-)). An affine space is a set of points and a vector space ##(M,V)##. Then you have a set of axioms which...

The theorem is as follows: All finite dimensional vector spaces of the same dimension are isomorphic Attempt: If T is a linear map defined as : T : V →W : dim(V) = dim(W) = x < ∞ & V,W are vector spaces It would be sufficient to prove T is a bijective linear map: let W := {wi}ni like wise let...

Homework Statement [/B] Which of the following sets are subspaces of ##R[x]?## ##W_1 = {f \in \mathbf R[x] : f(0) = 0}## ##W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}## ##W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}## ##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n...

Homework Statement Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V. Homework Equations W = {f ∈ V : f(1) = 1} W = {f ∈ V: f(1) = 0} W = {f ∈ V : ∃f ''(0)} W = {f ∈ V: ∃f ''(x) ∀x ∈ R} The...

I feel like the vector space ##\mathbb{R}^n## differs from other vector spaces, like ##\mathbb{P}##. For example, if we wrote down an element of ##\mathbb{P}##, like ##1+2t^2##, this is an object in its own right, with no reference to any coordinate system or basis. However, when I write down an...

Could we have two vector spaces each with its own set of basis vectors. but these basis vectors are related according to the following way. A particular set of vectors in the first vector space may exist "all over the place" but when you represent the same information in the second vector space...

Does it make sense to say that a set together with a field generates a vector space? I came across this question after starting the thread https://www.physicsforums.com/threads/determine-vector-subspace.941424/ To be more specific, suppose we have a set consisting of two elements ##A = \{x^2, x...

Hello Everyone. I am searching for some clarity on this points. Thanks for your help: Based on Schrodinger wave mechanics formulation of quantum mechanics, the states of a system are represented by wavefunctions (normalizable or not) and operators (the observables) by instructions i.e...

Hello, I think I understand what a vector space is. It is inhabited by objects called vectors that satisfy a certain number of properties. The vectors can be functions whose integral is not infinite, converging sequences, etc. The vector space can be finite dimensional or infinite dimensional...

Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. Homework Equations I know the 8 axioms about the vector spaces. The...

Hello, everybody! I would really appreciate if someone could help me understand how to solve the following two tasks. I am not sure whether my translation is correct, so if, by any chance, you know a more appropriate terminology, please let me know. I am not fluent in writing matrices here on...

I know that vector spaces have more structure as they are defined over fields and that modules are defined over rings. But it's hard to think of a situation where a using a ring clearly backfires. Is it just because a ring doesn't have an inverse for the second operation? For a module over Z...

Introduction Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can be something we use to count or more advanced an element of a field like real numbers. Students might answer that a number is a scalar. This is...

I am confused why is space over field ##R## not over field ##C## ? The entries in each vector is an element of ##\Bbb C## not ##\Bbb R##.

<Mentor's note: moved from a technical forum, therefore no template.> I'm long out of college and trying to teach myself QM out of Shankar's. I'm trying to understand the reasoning here because I think that I am missing something... 1.1.3 1) Do functions that vanish at the endpoints x=0 and...

Is the set of all differentiable functions ƒ:ℝ→ℝ such that ƒ'(0)=0 is a vector space over ℝ? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces...

In a vector space can you limit the scalar. For example, if I have Vector space in ℤ2 can i only multiple integer scalars?

Hello Forum, The state of a quantum system is indicated by##\Psi## in Dirac notation. Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis...

Problem: Let f ∶ V → V be a linear operator on a finite-dimensional vector space V . Prove that the sequence 0 → ker(f) → V → im(f) → 0 is exact at each term. Attempt: If I call: a: 0 → ker(f), b: ker(f) → V, c: V → im(f), d: im(f) → 0. Then the sequence is exact at: ker(f) if...

Hey! :o I want to check if the following are true. $V_1=\{a\in \mathbb{R}\mid a>0\}$ with the common multiplication as the vector addition and the scalar multiplication $\lambda \odot v=v^{\lambda}$ is a $\mathbb{R}$-vector space. $V_2=\{(x,y)\in \mathbb{Q}^2 \mid x^2=-y^2\}$ with the...

I need some help understanding one task. I know that for some structure to be a vector space all axioms should apply. So if any of those axioms fails then the given structure is not a vector space. Anyway, I have a task where I need to check if $$ \mathbb{C}^n_\mathbb{R} $$ is a vector space...

I searched for a proof of the statement in the title and found this document. But it just proves that for two norms ## \rho(x) ## and ## ||x|| ##, we have ## m\rho(x)\leq ||x|| \leq M \rho(x) ## for some m and M. But how does it imply that the two norms are equivalent? Thanks

I have some conceptual issues with functions in vectors spaces. I don't really get what are really the components of the vector / function. When we look at the inner product, it's very similar to dot product, as if each value of a function was a component : So I tend to think to f(t) as the...

Let ##(V, ||\cdot||)## be some finite-dimensional vector space over field ##\mathbb{F}## with ##\dim V = n##. Endowing this vector space with the metric topology, where the metric is induced by the norm, will ##V## become a topological vector space? It seems that this might be true, given that...

Typically an element of a vector space is called a vector, but Carroll's GR book repeatedly refers to elements of tangent spaces as "transforming as a vector" when they change coordinates as Vμ = ∂xμ/∂xν Vν. However, dual vectors are members of vector spaces (cotangent space) but obey ωμ =...

So I understand how to prove most of the axioms of a vector space except for axiom 10, I just do not understand how any set could fail the Scalar Identity axiom; Could anybody clarify how exactly a set could fail this as from what I know that anything times one results in itself 1u = u...

Homework Statement The question asks to show whether the following are sub-spaces of R^3. Here is the first problem. I want to make sure I'm on the right track. Problem: Show that W = {(x,y,z) : x,y,z ∈ ℝ; x = y + z} is a subspace of R^3. Homework Equations None The Attempt at a Solution...

During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture. Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent. ##W=\sum V_i## is a direct sum. Decomposition of...

Homework Statement Let U is the set of all polynomials u on field \mathbb F such that u(3)=u(-2)=0. Check if U is the subspace of the set of all polynomials P(x) on \mathbb F and if it is, determine the set W such that P(x)=U\oplus W. Homework Equations -Polynomial vector spaces -Subspaces...

Homework Statement Prove that \dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L) for every subspace \mathbb{F} and every linear transformation L of a vector space V of a finite dimension. Homework Equations -Fundamental subspaces -Vector spaces The Attempt at a Solution Theorem: [/B]If...

In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...

Homework Statement Show that finite dimensional normed vector spaces are complete. Homework Equations ##E## is a finite dimensional vector space and ##N## a norm on ##E## The Attempt at a Solution If ##\{x_n\}_n## is a Cauchy sequence in ##(E,N)##, then it is bounded and each term of the...

Cooperstein (in Advanced Linear Algebra) and Roman (also in a book called Advanced Linear Algebra) give versions of the Correspondence Theorem for Vector Spaces ... but these 'versions' do not look like the same theorem ... can someone please explain how/why these two versions are actually the...

Homework Statement Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.3 The Correspondence and Isomorphism Theorems ... ... I need further help with understanding Theorem 2.15 ... Theorem 2.15 and its proof read as follows...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.3 The Correspondence and Isomorphism Theorems ... ... I need help with understanding Theorem 2.15 ... Theorem 2.15 and its proof read as follows:It appears to me (and somewhat surprises me)...