Vector space Definition and 540 Threads

I am revising vector spaces and have got stuck on a problem that looks simple ... but ... no progress ... Can anyone help me get started on the following problem .. Determine the basis of the following subset of \mathbb{R}^3 : ... the plane 3x - 2y + 5z = 0 From memory (I studied vector...

Homework Statement Show that C[a; b], with the usual scalar multiplication and addition of functions, satises the eight axioms of a vector space. Homework Equations Eight Axioms of Vector Space: A1. x + y = y + z A2. (x+y)+z=x+(y+z) A3. There exists an element 0 such that x + 0 =...

Homework Statement Show that the set of twice differentiable functions f: R→R satisfying the differential equation sin(x)f"(x)+x^{2}f(x)=0is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f"...

Hello. I've just read about natural identifications of exterior powers with spaces of alternating maps, etc here: Some Natural Identifications However, I have problems showing that the following operations give the same space: V \rightarrow \Lambda_p V \rightarrow (\Lambda_p V)^* \cong...

Show that the sets \{a,b\} and \{a, b, a-b\} of real vectors generate the same vector space. How to proceed with it? I guess the following expression is helpful. c1*a+c2*b+c3*(a-b)=(c1+c3)*a+(c2-c3)*b=k1*a+k2*b

Hi everyone, :) Here's a question I am struggling with recently. Hope you can give me some hints or ideas on how to solve this. Question: If the collection of subspaces of the \(K\)-vector space \(V\) satisfies either distributive law \(A+(B\cap C)=(A+B)\cap (A+C)\) or \(A\cap (B+C)=(A\cap...

Hi everyone, :) Here's a problem that I need some help to continue. I would greatly appreciate if anybody could give me some hints as to how to solve this problem. Problem: Let \(f:V\rightarrow F\) be a linear function, \(f\neq 0\), on a vector space \(V\) over a field \(F\). Set...

Homework Statement Let V=Pol_3(R) be the vector space of polynomials of degree \leq3 with real entries. Let U be the subspace of all polynomials in V of the form aX^3+(b-a)X^2+bX+(d-b) and W be the subspace of all polynomials in V of the form aX^3+bX^2+cX+d such that a+c-d=0 (i) Does...

Hi, I'm trying to justify to myself the abstract notion of a vector space and I would really appreciate if people wouldn't mind taking a look at my description and letting me know if it's correct, and if not, what is the correct explanation? : "Vectors are most often introduced as ordered...

Let n be a positive integer, and for each $j = 1,..., n$ define the polynomial $f_j(x)$ by f_j(x) = $\prod_{i=1,i \ne j}^n(x-a_i)$ The factor $x−a_j$ is omitted, so $f_j$ has degree n-1 a) Prove that the set $f_1(x),...,f_n(x)$ is a basis of the vector space of all polynomials of degree ≤ n -...

Homework Statement The vector space V is equipped with a hermitian scalar product and an orthonormal basis e1, ..., en. A second orthonormal basis, e1', ..., en' is related to the first one by \mathbf{e}_j^{'}= \displaystyle\sum_{i=1}^n U_{ij}\mathbf{e}_i where Uij are complex numbers...

1. Determine whether W is a subspace of the vector space. W = {(x,y,z): x ≥ 0}, V = R3 I am not sure if I am doing this right. 2. Test for subspace. Let these conditions hold. 1. nonempty 2. closed under addition 3. closed under scalar multiplication 3. Testing for...

Homework Statement There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4. I need to find another basis B' for R4[z] such that no scalar multiple of an element in B appears as a basis vector in B' and also prove that...

Assume also that S + T = Iv and that S ∘ T = Ov = T ∘ S. Prove that V = X ⊕ Y where X = range(S) and Y = range(T). I don't understand how to go about it, please help.

Homework Statement Let V1 and V2 be vector spaces over the same field F. Let V = V1 X V2 = {f(v1, v2) : v1 \in V1; v2 \in V2}, and define addition and scalar multiplication as follows.  For (v1, v2) and (u1, u2) elements of V , define (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2).  For...

I've been thinking about a problem I made up. The solution may be trivial or very difficult as I have not given too much thought to it, but I can't think of an answer of the top of my head. Let ## T:V → V ## be a linear operator on a finite-dimensional vector space ##V##. Does there exist a...

So, I have an equivalence I need to prove, but I think I'm having trouble understanding the problem at a basic level. The problem is to prove that the inner product of a and b equals 1/4[|a+b|^2 - |a-b|^2] (a, b in C^n or an n-dimensional vector space with complex elements). I don't...

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms. 1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a...

Homework Statement Find a basis for the following vector space: The set of 2x2 matrices A such that CA=0 where C is the matrix : 1 2 3 6The Attempt at a Solution I multiplied C by a general 2x2 matrix ...

Its been a while since I've done this stuff, and I don't have a text handy. I know that for sets, intersection distributes over union, I don't remember if the same will hold for vector spaces over addition? for example does A \cap (B + C) = A \cap B + A \cap C

Suppose ##A## is a ## n \times n## matrix. Define the set ## V = \{ B | AB = BA, B \in M_{n \times n}( \mathbb{F}) \} ## I know that ##V## is a subspace of ##M_{n \times n}( \mathbb{F}) ## but how might I go about finding the dimension of ##V##? Is this even possible? It seems like an...

Homework Statement Prove that a0 = 0 Homework Equations The Attempt at a Solution Let V be a vector space on a field F. Let x be a member of V and a be a member of F. Consider that the 0 vector is the unique vector such that x + 0 = x Now, apply a scalar multiplication by a to both sides...

Hello, I'd like to make a, probably stupid, question regarding the axioms that define a vetor space. Among them, there are the axioms: λ\cdot(μ\cdotX) = (λμ)\cdotΧ (1) and 1\cdotΧ=Χ (2) for all λ,μ in the field and for all X in the vector space, where 1 is the identity of the...

Consider the operation of multiplying a vector in ℝ^{n} by an m \times n matrix A. This can be viewed as a linear transformation from ℝ^{n} to ℝ^{m}. Since matrices under matrix addition and multiplication by a scalar form a vector space, we can define a "vector space of linear transformations"...

I'm asking because I think of Minkowski space as a manifold with a Riemannian metric. However, I've also seen treatments in which an event in spacetime is chosen as origin, and special relativity treated as a vector space given the choice of origin. Does this mean that a vector space is a...

Homework Statement Is the set of all polynomials with positive coefficients a vector space? It's not. But after going through the vector space conditions I don't see how it can't be.

I have a fixed function U and a function f that I want to know something about, both from \mathbb{R}^{n}\rightarrow\mathbb{R}, for which I know condition (G) that U(x)\geq{}U(y) implies f(x)\geq{}f(y). f=U and f=x_{0} for any x_{0}\in{}\mathbb{R} trivially fulfill (G). Which f fulfills (G)...

Homework Statement Let V:= ℝ_{2}[t] V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t) a) Check that f \in End(V) b) Calculate the characteristic polynomial of f. Homework Equations The Attempt at a Solution a) Is it sufficient to check that (f+g)(t)=f(t)+g(t) ...

Homework Statement Consider the space of continuous functions in [0,1] (that is C([0,1]) over the complex numbers with the following scalar product: ##\langle f , g \rangle = \int _0 ^1 \overline{f(x)}g(x)dx##. Show that this space is not complete and therefore is not a Hilbert space. Hint:Find...

Homework Statement V' is the adjoint space of the vector space V. For a mathematician, what objects comprise V'? Homework Equations The Attempt at a Solution V' comprises functions, which when applied on the elements of V, produce complex scalars.

Homework Statement 1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False? 2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all...

Homework Statement What properties cause complete sets of amplitudes to constitute the elements of a vector space? Homework Equations The Attempt at a Solution Does the question mean 'a vector space' or 'a linear vector space'?

Hello All, I was trying to prove that an operator T in a real vector space V has an upper block triangular matrix with each block being 1 X 1 or 2 X 2 and without using induction. The procedure which i followed was : We already know that an operator in a real vector space has either a one...

Homework Statement Determine whether this set equipped with the given operations is a vector space. For those that are not vector spaces identify the axiom that fails. Set = V = all pairs of real real numbers of the form (x,y) where x>=0, with the standard operations on R^2. The...

Homework Statement Let V be the set of all nonconstant functions with operations of pointwise addition and scalar multiplication, having the real numbers as their domain. Is V a vectorspace? Homework Equations None. The Attempt at a Solution My guess is, no. For example F(x) =...

So I just got out of my linear algebra midterm, and this question is confusing the hell out of me. Basically, it's a subspace of R^4, such that the coordinates satisfy the following qualifications: (a, b - a, b, 2(b - a)) So basically, a and b can range over the xz plane, and y and w sort...

I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...

Homework Statement The set of all real-valued functions f defined everywhere on the real line and such that f(4) = 0, with the operations (f+g)(x)=f(x)+g(x) and (kf)(x)=kf(x) verify if all axioms hold true. Homework Equations Axioms 1 and 6 These closure axioms require that if we add...

Here is the question: Here is a link to the question: Proving a set V is a vector Space? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Let I = [a,b], a closed interval. With addition and scalar multiplication as defined for all real-valued continuous functions defined on I, determine which of the following sets of functions is a vector space. a) All continuous functions, f, such that f(a)=f(b) b)...

I'm trying to understand this problem. Let's take an infinite dimensional vector space, say \mathbb{R}^2 and let n = 4. This problem states we can find a subspace $U$ such that dim(\mathbb{R}^2/U) = 4$. Well, one subspace of $\mathbb{R}^2$ is U = \{(x, y) : x*y \geq 0\} (i.e. the first and third...

Homework Statement A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that (a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k (b) the one sided limits exist as finite numbers Let V be the...

In the book it states that the span of the empty set is the trivial set because a linear combination of no vectors is said to be the 0 vector. I really don't know how they came up with at? Is it just defined to be like that? After doing some research, I figured that since the empty set is a...

I'm unable to understand this generalization of vectors from a quality having a magnitude and direction, to the more mathematical approach. what is the difference between vector space and vector field? more of an intuitive example?

Hi, I am currently reading about differential forms in "Introduction to Smooth Manifolds" by J. M. Lee, and I was wondering exactly how you define the wedge product on the exterior algebra \Lambda^*(V) = \oplus_{k=0}^n\Lambda^k(V) of a vector space V. I understand how the wedge product is...

Hi, Algebraists: Say V is finite-dimensional over F . Is there more than one way of defining the action of F on V (of course, satisfying the vector space axioms.) By different ways, I mean that the two actions are not equivariant. Thanks.

Homework Statement Hi guys, I'm trying to prove that matrix inversion is continuous. In other words, I'm trying to show that in a normed vector space the map \varphi: GL(n,R) \to GL(n,R) defined by \varphi(A) = A^{-1} is continuous.Homework Equations The norm that we're working in the...

Guys I am having a little trouble understanding how and why we use complex vector spaces to describe the quantum states of a particle. Why complex vector spaces, and how is a complex vector space defined. Also are the 'vectors' in the field of quantum mechanics simply elements of a vector...

Prove that \(R^{\infty}\) is infinite dimensional.

Homework Statement Suppose B = {u1, u2.. un} is a basis of V. Let U = {u1, u2...ui} and W = {ui+1, ui+2... un}. Prove that V = U ⊕ W. Homework Equations The Attempt at a Solution I think I should prove that elements in U are not in W and viceversa. Then this prove it is indeed a...