Vector space Definition and 530 Threads

I'm learning about rings, fields, vector spaces and so forth. The book I have states: "Real-valued functions on R^n, denoted F(R^n), form a vector space over R. The vectors can be thought of as functions of n arguments, f(x) = f(x_1, x_2, ... x_n) "It then says later that these vectors...

This is an extract from my third year 'Foundations of QM' lecture notes: If ψ1 and ψ2 are admissible states, then the superposed state \alphaψ1 + ψ2\beta ( \alpha,\beta \in C ) is also an admissible state. \rightarrow complex vector space. I understand that a linear superposition...

Given a measurable function f that is not real- or complex valued, but that maps into some vector space, what are the necessary conditions for it to be integrable? I've looked through over 20 books on integration and measure theory, but they all only deal with integration of real (or...

The finite case is fine, as a vector space it is easy to show that R^X is isomorphic to R^n. What about when X is infinite? I believe it is true in general that dim(R^X) = #(X), which I hope holds in the infinite case too. I know that the set given by B={b_x; x in X} defined as b_x(y) =...

Homework Statement Let P denote the set of all polynomials whose degree is exactly 2. Is P a vector space? Justify your answer. Homework Equations (the numbers next to the a's are substripts P is defined as ---->A(0)+A(1)x+A(2)x^2 The Attempt at a Solution I really don't...

Homework Statement If we have a normed vector space, and a sequence of vectors \{\mathbf{v}_k\}_{k=1}^{N} in the normed vector space. If there exists a constant B>0 such that the following holds for all scalar coefficients c_1,c_2\cdots c_N B\sum\limits_{k=1}^N |c_k|^2 \leq...

I'm currently doing a self study course on Linear Algebra. Can anyone give me an example of vector space and basis with reference to Structural Engineering? For example I have a displacement vector for a simply supported beam as: [thata_a theta_b]^T where; theta_a and theta_b...

Homework Statement http://img267.imageshack.us/img267/8924/screenshot20120118at121.png The Attempt at a SolutionWe have that X = A + B. To show that X is unique, let two such sums be denoted by X1 X2 such that X1 ≠ X2. We write, X1 = A + B X2 = A + B The equations imply, X1 - A - B = 0 X2...

Homework Statement Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms. Homework Equations N/A The Attempt at a Solution I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove...

I was thinking this: If I have the set A = \{ \mathbb{R}^n, \mathbb{R}^n \} where for the "first" element I mean the real vector space \mathbb{R}^n , and the "second" element is the additive group \mathbb{R}^n , then does the set A contain one element ( \mathbb{R}^n )? Or it contains two...

Homework Statement Let S be any non-empty set, F be a field and V={ f : S -> F such that f(x) = 0 } be a vector space over F. Let f[sub k] (x) : S -> F such that f[sub k] (x) = 1 for k=x, otherwise f[sub k] (x) = 0. Prove that the set { f [sub k] } with k from S is a basis for the vector space...

Homework Statement Show M_mn(F) (the collection of mxn matrices over F) over a field F is a vector space. The Attempt at a Solution Denote A=a_{ij},B=b_{ij} for elements of M_{mn}(F) . Define A+B=(a_{ij}+b_{ij}) and for a\in F denote \alpha A=(\alpha a_{ij}). Then, (a) If...

i need to a buk that explains linear vector space with respect to quantum mechanical concepts...most buk have just a short review or something..i need to understand it and how it is related to qm... thanks a lot

Homework Statement I was wondering if someone could explain the easiest way to determine if a set S spans V? some example questions would be: show that S = {v1, v2, v3, v4} spans R4 where v1 = [1 0 +1 0] v2 = [0 1 -1 2] v3 = [0 2 +2 1] v4 = [1 0 0 1] Homework Equations The...

Homework Statement For x, y in a vector space V, c in F, if cx=0 then c=0? How do you prove this? This is originally from Friedman, Linear Algebra, p.12. To prove this, I can use a few facts: (1) cancellation law (2) 0, -x are unique (3) 0x = 0 with basic vector space definition...

Hi guys, I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars) However, then I was asked...

Hi all, it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product. For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}. However, does this...

Homework Statement This is only part of a problem I am working on, but the only part that I have questions about is the following: Show that \mathcal{F}(\mathbb{R}) is infinite dimensional. Homework Equations \mathcal{F}(\mathbb{R}) is the set of all functions that map real numbers to real...

Let V= \mathbb{R}_3[x] be the vector space of polynomials with real coefficients with degree at most 3 and let D:V\to V be the linear operator of taking derivatives, D(f)=f'. I'm trying to check the Rank-nullity theorem for this example but it doesn't seem to hold: Since D is not injective...

Homework Statement Let V be the set of ordered pairs (x, y) of real numbers with the operations of vector addition and scalar multiplication given by: (x, y) + (x', y') = (y + y', x + x') c(x, y) = (cx, cy) V is not a vector space. List one of the properties from the definition of vector...

Prove that the space of 2×2 real matrices forms a vector space of dimension 4 over R. [12] Im unsure, anyone any idea?

Homework Statement Find a basis for (1, a, a^2) (1, b, b^2) (1, c, c^2) Homework Equations The Attempt at a Solution M(1, a, a^2) + N(1, b, b^2) + K(1, c, c^2) = (0, 0, 0) M + N + K = 0 Ma + Nb + Kc = 0 Ma^2 + Nb^2 + Kc^2 = 0 This is as far as I got. I tried monkeying around with these 3...

I looked up what is a vector space online and it always gives like formula or long explanations. In a couple sentences can you tell me exactly what a vector space is? I know it has to go through the origin, but what else is true about a vector space?

Hey guys, this question is more or less related to the way Frobenius' theorem is presented in my text. Consider an n - manifold M, an m - dimensional submanifold S of M, and a set of k linearly independent vector fields V^{\mu }_{(a)} such that k \geq m. In order for S to be an integral...

Homework Statement Let V = C (complex numbers). Prove that the only C-subspaces of V are V itself and {0}. Homework Equations The Attempt at a Solution Well this problem has me confused since I have clearly found a complex subspace for example all the complex numbers of the form {a+ib ...

All continuous functions on closed interval [a, b] form a vector space. The functions in this space are the vectors. However what is the physical significance of the norm of a vector in this space? For example if we found the norm of a function is 1/3 what does this signify? Does it dependent on...

Homework Statement Denote the inner product of f,g \in H by <f,g> \in R where H is some(real-valued) vector space a) Explain linearity of the inner product with respect to f,g. Define orthogonality. b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be...

I know that a set (let's call it V) of all functions which map (R -> R) is a vector space under the usual multiplication and addition of real numbers, but i am having trouble proving it, i understand that the zero vector is f(x)=0, do i just have to prove that each element of V remains in V...

hi friends ! it is well known that a Lie algebra over K is a K-vector space g equipped of a K-bilinear, called Lie bracket. I ask how can we determines the Lie algebra of any vector space then? For example we try the Lie algebra of horizontal space.

Homework Statement Let F be a field. Prove that the set of polynomials having coefficients from F and degree less than n is a vector space over F of dimension n. Homework Equations The Attempt at a Solution Since the coefficients are from the field F, the are nonzero. So, if...

The book I use for linear algebra explains that the motivation for defining a vector space has to do with the Gauss' reduction method taking linear combinations of the rows, but I don't understand the explanation very well. Can somebody explain?

I have a question: If x\in X is a normed vector space, X^* is the space of bounded linear functionals on X, and f(x) = 0 for every f\in X^*, is it true that x = 0? I'm reasonably confident this has to be the case, but why? The converse is obviously true, but I don't see why there couldn't be an...

hi, anyone can provide a simple explanation of what is a dual vector space? i have scoured the net and the explanations are all a tad too complicated for my understanding :( thanks

A non-zero alternating tensor w splits the bases of V into two disjoint groups, those with \omega(v_1,\cdots,v_n)>0 and those for which \omega(v_1,\cdots,v_n)<0. So when we speak of the orientation of a vector space, we need to say the orientation with respect to a certain tensor, correct?

Given two orthonormal bases v_1,v_2,\cdots,v_n and u_1,u_2,\cdots,u_n for a vector space V, we know the following formula holds for an alternating tensor f: f(u_1,u_2,\cdots,u_n)=\det(A)f(v_1,v_2,\cdots,v_n) where A is the orthogonal matrix that changes one orthonormal basis to another...

It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded). Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous? Thanks

How do I show the vectors, polynomials, and matrices generate the given sets? A subset of a vector space generates the vector space if the span of the subset is the vector space. The span is the set of all linear combinations. For 6, do I show the vector are linearly independent and can thus...

Hi all, It has been very useful of posting my questions here to help me pushing through the book reading of analysis. This forum is a perfect place and the best place for people who are interested in knowledge and the beauty of knowledge. Here goes another question from me. All continuous...

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110606_200153.jpg The book says x + y = 0 is not satisfied. That is, for each x in V, there is a y that satisfies the equation, which is an additive inverse. How vector addition is defined, for each x, I could simply set b1 = -a1 and...

Hi, :) I have some basic knowledge of matrices and vectors. mI understand some of their practical applications in real life. This idea of vector is entirely new to me. What is a vector in simpl terms, or to begin with? Please try to explain with some simple practical explanation. I would...

I am using Axler's Linear Algebra Done Right as a text for independent study of linear algebra. Axler basically defined a vector space to be a set which has defined operations of addition and multiplication (and which comports with certain algebraic properties) and that contains an additive...

Homework Statement Having a symmetric tensor S^{a_1 ...a_n} forming a vector space V_n with indices taking values from 1 to 3; what is the dimension of such a vector space? Homework Equations The Attempt at a Solution essentially this reduces to picking a tensor of type S^{...

Homework Statement A spin-1 particle is measured in a stern gerlach device, set up to measure S_{z}. What are the possible outcomes? In this case, the outcome is zero. The same particle is measured by a second deviced which measures S_{x}. What are the possible outcomes of this...

Homework Statement A matrix a is idempotent if a^2=a. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents 2. The attempt at a solution the vector space in question is dimension 4, so I need to find 4 idempotent matrices. but i don't want to find them...

"Usefulness" of Basis for a Vector Space, General? Hi, Everyone: I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension" a student asked me what was the use or purpose of a basis for a vector space V. All I could think of is that bases allow us to...

vector space? Let v denote the set of order pairs of real numbers. If(a1,a2) and (b1,b2) are elements of V and c is an element of the reals, define (a1,a2)+(b1,b2)=(a1+b1,a2b2) and c(a1,a2)=(ca1,a2) is v a vector space over reals with these operations? im thinking its not because the...

Homework Statement Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. "The set of all 2 x 2 singular matrices with the standard operations." Homework Equations...

Homework Statement Let S be a subspace of a vector space V. Let B be a basis for V. Is there a basis C for S such that C \subseteq B? not really sure how to approach this... any hints?

I am confused as to exactly what a local base at zero (l.b.z.) tells us about a topology. The definition given in Rudin is the following: "An l.b.z. is a collection G of open sets containing zero such that if O is any open set containing zero, there is an element of G contained in O". Ok, great...

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\}...