Vector space Definition and 540 Threads

Let V be a vector space over an infinite field $\mathbf{k}$. Let \beta be a basis of V. In this case we can write V\cong \mathbf{k}^{\oplus \beta}:=\bigl\{ f\colon\beta\to \mathbf{k}\bigm| f(\mathbf{b})=\mathbf{0}\text{ for all but finitely many }\mathbf{b}\in\beta\bigr\}...

1. Homework Statement [PLAIN]http://img87.imageshack.us/img87/9001/150674.png 2. Homework Equations 3. The Attempt at a Solution I think, I need to write: a(x)[b(+)c] Then prove: Distributivity law 1: Unsure of Distributivity law 2: Proving: a(x)[b(+)c] =...

Homework Statement Show that all 2 x 2 matrices with real entries: M(2x2) = { a b | a,b,c,d are real numbers} c d | is a vector space under the matrix addition: |a1 b1| + | a2 b2| = |a1+a2 b1+b2| |c1 d1| + | c2 d2| = |c1+c2 d1+d2| and scalar multiplication: r*| a b | = | ra...

Let G be the set of all real functions f(x) each of which is analytic in the interval [0,1] and satisfies the conditions: f(0)+a*f'(0)=0; f(1)+b*f'(1)=0, where (a,b) is a pair of real numbers from the set D={(a,b) in R^2: 1+b-a!=0 (is not equal to zero)}. Prove that the set G is a linear vector...

Homework Statement If R3 is a vector space and V = (x,x,0) is a subspace, find unique subspaces W1 and W2 such that R3 = V ⨁ W1 = V ⨁ W2 Homework Equations The Attempt at a Solution Assuming R3 = (x,y,z) - please correct me if I'm wrong somehow - then I could pick a W1 like...

Homework Statement Determine whether the following set forms a vector space: {(x1, x2, x3) E R^3 | x1 + 2x2 - x3 = 0 and x1x2 = 0} Homework Equations The axioms! The Attempt at a Solution I know that the first equation in the set fulfills the axioms for a vector space, since...

If you have a vector space you can find a set of elements and consider their span, and then look for elements that cannot be spanned by them and so add them to the set, if you can't add anymore then you have a basis. My question is what happens if this process continues forever, do you...

I'm still confused as to what a vector space is. Is it just some set of numbers that meet the conditions to be closed? So my main problem: V is the set of all polynomials of the form at2+bt+c, where a,b,c are real numbers wtih b=a + 1 (a1t2+b1t+c1) + (a2t2+b2t+c2) =...

Just for clarification, if a vector space is defined over a field F, are entries inside the vectors in the vector space necessarily restricted to field F? Say I had a vector space V={(a1,a2,...):ai∈C} , could the vector space be over the field R so that I only take scalars from the reals?

Hi all. First of all I want to say that I am new here and I want to apologise if this is the wrong forum to post my question. :) I've just finished a sort of a crash course on functional analysis that left much more questions in me than it answered but one thing bothers me a lot. When we talk...

Homework Statement Consider the vector space C[a,b] of all continuous complex-valued functions f(x), x \in [a,b]. Define a norm ||f|| sup = max{|f(x)|, x \in [a,b]}. (Math Note: technically we want to use sup instead of max but a physicists operational definition of max is the mathematial...

Homework Statement Is the following a vector space: The set of all periodic functions of period 1? ( i.e. f(x+1)=f(x) ) Homework Equations If v1 and v2 are in V then v1 + v2 is in V If v1 is in V then c*v1 is in V where c is a scalar The Attempt at a Solution I'm thinking no...

Homework Statement Prove: The set S(V) of all subspaces of a vector space V is a complete lattice under set inclusion, with smallest element {0}, largest element V, meet glb(S_{i} | i \in K) = \cap_{i \in K} S_{i} and join lub(S_{i} | i \in K) = \sum_{i \in K} S_{i} (Btw, how can I write...

You know the example "The space of functions from a set S to a field F" that's usually given in a linear algebra text? Well they never give an example of the set they're working in in detail so I defined the set as: ((S, (S x S, S, +)), ((F, (F x F, F, +')), (F x F, F, °)), (S x F, F, •))...

Homework Statement How can I find the base and dim of U here?, V = P3; U = {p in P3 : p'(0) = p(1)}... Homework Equations The Attempt at a Solution now I've proven it is a subspace and that it is closed under addition and scalar multiplication...but how can I find the base and...

Homework Statement I need to prove that, if {u;v;w} is a linearly independent set in a vector space, then the set {2u + v + w; u + 2v + w; u + v + 2w} is also linearly independent. Homework Equations ... The Attempt at a Solution if {u;v;w} is a linearly independent set=>...

Homework Statement I'm trying to define a vector space over Q. Does this make any sense? Homework Equations The properties of a vector space The Attempt at a Solution Let V=Q^2 over Q. It seems to me that everything would be defined and I shouldn't be able to do anything to a...

Let W be a subspace of a vector space V. We define a relation v~w if v-w is an element of W. It can be shown that ~ is an equivalence relation on V. Suppose that V is R^2. Say W1 is a representative of the equivalence class that includes (1,0). Say W2 is a representative of the equivalence...

I was given a set of problems and answers, but not the solutions. This is one of the questions I've been having trouble in getting the correct answer. Homework Statement Determine whether the set of vectors (u, v) is a vector space, where u + v = 0. Homework Equations The 10 axioms of...

Hi, I'm having trouble understanding a proof of the following theorem which allows it to be shown that all bases for a vector space have the same number of vectors, and that number is the dimension of the vector space (as you probably already know): Homework Statement Theorem: If S =...

Homework Statement Q) Determine if the set is a vector space under the given operations The set of all pairs of real numbers of the form (1,x) with the operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky) 2. Homework Equations / Solution Attempt I know the axioms needed in this...

So I'm pretty sure I understand the formalism of dual vector spaces. (E.g. there exist objects that operate on vectors and take them to scalars. these objects themselves form a linear vector space). But I'm having difficulty understanding where this comes from intuitively. How would I know...

A vector space is a collection of vectors. So can we say that it is a set (although with special properties) ? Just wanted to confirm this. Is this also true for a vector space consisting of all functions from R to R , i.e can we say that it is also a set , where each member of the set is a...

Can some one explain why not all norms are equivalent in and infinite vector space? Examples/Counter examples? How would you go about proving/disproving this? Any or all of the above or just any help anyone has to offer would be great Thanks

b]1. Homework Statement [/b] Let E be a normed vector space. Let (x_n) be a convergent sequence on E and x its limit. Prove that A = {x}U{x_n : n natural number} is compact. Homework Equations A is compact iff for any sequence of A, it has a cluster point, say a in A, i.e. there is a...

Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...

Homework Statement Given is a vector space (V,+,k) over kn with n > 1. Show that with W \subseteq V, U \subset V and dim(U) = n - 1 dim(W \cap U) \geq dim(W) - 1 Homework Equations dim(W+U) +dim(W \cap U) = dim(W) +dim(V) The Attempt at a Solutiondim(V) = n dim(W) \leq dim(V) dim(W+U)...

Homework Statement show whether the following set of vectors M = \left\{\left(a_{1},a_{2},a_{3}\right) with a_{1},a_{2},a_{3} \in \Re\right\} with the following limitations: 1) a1 is rational 2) a1 = 0 3) a1 + a2 = 0 4) a1 + a2 = 1 is a vector space over the field of real numbers. Homework...

Can you guys provide a counter example of why this statement is False. If T: V-->V with V a VS over C then T has an eigenvector? This is not always true as if V is infinite dim., it'll have a Spectrum. Any counter examples? Thanks

if we have a vector space V,can we define the tangent bundle of V rated TV?

Homework Statement The set of all 2x2 singular matrices is not a vector space. why? \begin{bmatrix} 1 & 0\\ 0&0 \end{bmatrix}+\begin{bmatrix} 0 & 1\\ 0& 1 \end{bmatrix}=\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} Homework Equations Is it because the determinant in both are zero, but by...

Homework Statement How many two dimensional subspaces does (F_3)^4 have? The attempt at a solution I chose an arbitrary basis so B = (v1,v2,v3,v4) for (F_3)^4 and then basically did 4C2 = 6 so it has 6 subspaces with dimension 2. However, thinking over this problem I've realized that I'm...

Is this a legitimate definition for an "absorbing set" in a topological vector space? A set A\subset X is absorbing if X = \bigcup_{n\in \mathbb N} nA. This is the definition the way it was presented to us in my functional analysis class, but I'm looking at other sources, and it seems everyone...

Is this a vector space? Let M2 denote the set of all matrices of 2 x 2. Determine if M2 is a vector space when considered with the standard addition of vectors, but with scalar multiplication given by α*(a b) = (αa b) (c d) (c αd) In case M2 fails to be a vector space with these...

Why does a vector space need to go through the origin? I am not sure I understand the geometric importance of that, or the importance of it as it pertains to the axioms of a vector space. Is this to say I want any vector in the vector space to lie completely in the space, as oppose to say...

Homework Statement We have a vector space (V, R, +, *) (R being Real numbers, sorry I couldn't get latex work..) with basis V = span( v1,v2). We also have bijection f: R² -> V, such as f(x,y) = x*v1+y*v2. Assume you have inner-product ( . , . ): V x V -> R. ( you can use it abstractly and...

If k is an integer between 0 and min(m,n),show that the set of mxn matrices whose rank is at least k is an open submanifold of M(mxn, R).Show that this is not true if "at least k"is replaced by "equal to k." For this problem, I don't understand why the statement is not true if we replace "at...

Hi, could you please tell me the collection of 3- vectors [a,b,c] form a vector space such that c - a =2b . Could you please explain it? thank you so much.

Can the hyperreal numbers be described as a vector space over the reals with a basis (0,...,e2,e1,e0,e-1,e-2...), where e1 is a first order infinitesimal number, e0 = 1, and e1 a first order infinite number?

In dealing with Poisson, Laplace, Schrodinger and other wave equations, one has to deal with propagating and evanescent waves. We know all about the propagating waves - orthogonality and completeness relations, but what about evanescent waves? Do they form a vector space with corresponding...

suppose we have a principle fiber bundle P at a point p \in P we have the decomposition T_pP=V_pP + H_pP it is said that the vertical subspace V_pP is uniquely defined while H_pP is not i cannot understand this point the complement to a unique subspace is surely unique, i think. it is a...

I'm not sure I fully understand the difference between these two terms when used in differential geometry/general relativity. If I were to describe covariant differentiation to someone, I would say something like this: "On a curved manifold (imagine a basketball), you could assume a tangent...

Homework Statement I uploaded the problem because its easier to see. Homework Equations The Attempt at a Solution The solution is there from a practice exam. I don't understand the x,y,z part and how it fulfills the multiplication requirement of a vector space. Could someone...

Homework Statement Let V be a vector space over the field K. a) Let {W_{k}:\ 1\leq k \leq m} be m subspaces of V, and let W be the intersection of these m subspaces. Prove that W is a subspace of V. b) Let S be any set of vectors in V, and let W be the intersection of all subspaces of V which...

Homework Statement Let V be a nonzero vector space over a field F, and suppose that S is a bases for V. Let C(S,F) denote the vector space of all functions f ∈ Ω(S,F) (i.e. the set of functions from S to a field F) such that f(s) = 0 for all but a finite number of vectors in S. Let Ψ: C(S,F)...

Homework Statement Let V be a nonzero vector space over a field F, and suppose that S is a bases for V. Let C(S,F) denote the vector space of all functions f ∈ Ω(S,F) (i.e. the set of functions from S to a field F) such that f(s) = 0 for all but a finite number of vectors in S. Let Ψ: C(S,F)...

Tensor product of vector space problms Homework Statement I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it...

I don't really understand what the dimension of a vector space for planes is. Is it 2 or 3 and why? What's the difference between the dimension of the plane as a surface (dimension of all surfaces is 2) and the dimension of plane in a vector space? Also, Wikipedia says that only planes passing...

Homework Statement Use the subspace theorem to decide which of the following are real vector spaces with the usual operations. a) The set of all real polynomials of any degree. b) The set of real polynomials of degree \leq n c) The set of real polynomails of degree exactly n...

Are functionals united with the vector space which they operate on? For example, Physics is a functional of Behavioral Psychology. However, Behavioral Psychology does not include Physics. Am I correct? Thank you, Jake