# Vector space Definition and 40 Discussions

A vector space (also called a linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms (listed below in § Definition). To specify that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.
Certain sets of Euclidean vectors are common examples of a vector space. They represent physical quantities such as forces, where any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same way (but in a more geometric sense), vectors representing displacements in the plane or three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.
Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis as function spaces, whose vectors are functions. These vector spaces are generally endowed with some additional structure such as a topology, which allows the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used (being equipped with a notion of distance between two vectors). This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis.
Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.
Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations. They offer a framework for Fourier expansion, which is employed in image compression routines, and they provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

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1. ### Show a function is linear

I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
2. ### Prove that ##S## is a subspace of ##V##

Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##? Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...
3. ### Prove that the inner product converges

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...
4. ### Help with linear algebra: vectorspace and subspace

So the reason why I'm struggling with both of the problems is because I find vector spaces and subspaces hard to understand. I have read a lot, but I'm still confussed about these tasks. 1. So for problem 1, I can first tell you what I know about subspaces. I understand that a subspace is a...
5. ### Linear algebra inner products, self adjoint operator,unitary operation

b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two. Otherwise does b-d above look correct? Thanks in advance!
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### I Proving linear independence of two functions in a vector space

Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
7. ### Determining whether a set is a vector space

Summary:: the set of arrays of real numbers (a11, a21, a12, a22), addition and scalar multiplication defined by ; determine whether the set is a vector space; associative law Question: determine whether the set is a vector space. The answer in the solution books I found online says that...
8. ### How to show a subspace must be all of a vector space

Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of ##R^2##)...
9. ### A Linearly independent function sets

It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...
10. ### I Expanding a given vector into another orthonormal basis

Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis. In equation 9.2.32, the basis of ## |V \rangle## is not given. ## |1 \rangle ## and ## |2 \rangle ## themselves form basis vectors. Then how can one calculate ## \langle 1| V \rangle ## ? Do...
11. ### Checking the linear independence of elements of 2 X 2 matrices

Homework Statement Homework Equations 3. The Attempt at a Solution [/B] ## |3 \rangle = |1 \rangle - 2 ~ |2 \rangle ## So, they are not linearly independent. One way to find the coefficients is : ## |3 \rangle = a~ |1 \rangle +b~ |2 \rangle ## ...(1) And solve (1) to get the values of a...
12. ### Functions forming a vector space

Homework Statement 1.1.3 1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space? 2) How about periodic functions? obeying f(0)=f(L) ? 3) How about functions that obey f(0)=4 ? If the functions do not qualify, list what go wrong. Homework Equations The Attempt at a...
13. ### [Linear Algebra] Show that H ∩ K is a subspace of V

Homework Statement From Linear Algebra and Its Applications, 5th Edition, David Lay Chapter 4, Section 1, Question 32 Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
14. ### I A different way to express the span

Let us assume that d is a vector in the vector space ℝ2 , then is: {td | t ∈ ℝ} the same as span{d} ? Thank you.
15. ### Vector Spaces Question

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...
16. ### Matrices and Linear spaces

Homework Statement The vectors ##a_1, a_2, a_3, b_1, b_2, b_3## are given below $$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1)$$ The subspace of ## \mathbb R^4 ## spanned by ##a_1, a_2, a_3## is denoted by...
17. ### Decomposing space of 2x2 matrices over the reals

Homework Statement Consider the subspace $$W:=\Bigl \{ \begin{bmatrix} a & b \\ b & a \end{bmatrix} : a,b \in \mathbb{R}\Bigr \}$$ of $$\mathbb{M}^2(\mathbb{R}).$$ I have a few questions about how this can be decomposed. 1) Is there a subspace $$V$$ of...
18. ### Complex periodic functions in a vector space

Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t) Homework Equations The Attempt at a Solution f(t) = e(i*w0*t)) g(t)...
19. ### Linear Algebra - Finding coordinates of a set

Homework Statement Find the coordinates of each member of set S relative to B. B = {1, cos(x), cos2(x), cos3(x), cos4(x), cos5(x)} S = {1, cos(x), cos(2x), cos(3x), cos(4x), cos(5x)} I am to do this using Mathematica software. Each spanning equation will need to be sampled at six separate...
20. ### Basis of the intersection of two spaces

Homework Statement Consider two vector spaces ##A=span\{(1,1,0),(0,2,0)\}## and ##B=\{(x,y,z)\in\mathbb{R}^3 s.t. x-y=0\}##. Find a basis of ##A\cap B##. I get the solution but I also inferred it without all the calculations. Is my reasoning correct Homework Equations linear dependence...
21. ### I What is a spinor?

Okay, I have read on spinors here and there but I really don't understand geometrically or intuitively what it is. Can someone please explain it to me and how/when it is used? Thanks!
22. M

### I Reflexive relation question

Hello all. I have a question about a reflexive relation. Consider ##1_V : V \rightarrow V## with ##V## a vector space. Obviously, this is an isomorphism. My book uses this example to show that V is isomorphic with V (reflexive relationship). However, suppose I have a function ##f: V\rightarrow...
23. M

### I Proof that every basis has the same cardinality

Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinality So, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...
24. M

### Vector space property

Homework Statement Prove that in any vector space V, we have: ##\alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0 \lor \overrightarrow a = \overrightarrow 0## Homework Equations I already proved: ##\alpha \overrightarrow 0 = \overrightarrow 0## ##0 \overrightarrow a =...
25. M

### Vector space properties: distributivity

Homework Statement I want to proof, using the axioms of a vector space, that: ##(\alpha - \beta)\overrightarrow a = \alpha \overrightarrow a - \beta \overrightarrow a## Homework Equations Definition vector space: The Attempt at a Solution ##(\alpha - \beta)\overrightarrow a = (\alpha +...
26. ### Distance from a point to a plane

Homework Statement What is the distance from the point P to the plane S? Homework Equations ## S = \left \{ r_{0} + s(u_{1},u_{2},u_{3})+t(v_{1},v_{2},v_{3}) | s,t \in \mathbb{R} \right \} ## The Attempt at a Solution [/B] I found an expression for the general distance between point P and a...
27. ### I Representation of a vector

Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...
28. ### Subspace question

Homework Statement Let X=ℝ3 and let V={(a,b,c) such that a2+b2=c2}. Is V a subspace of X? If so, what dimensions? Homework Equations A vector space V exists over a field F if V is an abelian group under addition, and if for each a ∈ F and v ∈ V, there is an element av ∈ V such that all of...
29. ### A Hyperhermitian inner product -- explicit examples requested

Can anyone show me explicit examples of Hyperhermitian inner product?
30. ### I What is the Poynting "vector" mathematically?

Hi. The Poynting vector is a 3-tuple of real or complex numbers (depending on the respective formulation of electrodynamics) times a unit. It may be pictured as an arrow with some length and direction in IR^3 or IC^3. But is it a "vector" in the strict mathematical sense, i.e. an element of a...
31. ### Matrix of linear transformation

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...
32. ### Linear system of equations

Homework Statement Plot the solution set of linear equations x-y+2z-t=1 2x-3y-z+t=-1 x+7z=8 and check if the set is a vector space. 2. The attempt at a solution Augmented matrix of the system: \begin{bmatrix} 1 & -1 & 2 & -1 & 1 \\ 2 & -3 & -1 & 1 & -1 \\ 1 & 0 & 7 & 0 & 8 \\...
33. ### B About dimension of vector space

I'm confused about this. I know that if the dimension of the vector space is say, 2, then there will be 2 elements, right? eg. ## \left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)## What I want to know is if the dimension of vector space is still two if the matrix is like this...
34. ### Direct Sum: Vector Spaces

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...
35. ### How to determine the smallest subspace?

Two examples are: Given two subspaces ##U## and ##W## in a vector space ##V##, the smallest subspace in ##V## containing those two subspaces mentioned in the beginning is the sum between them, ##U+W##. The smallest subspace containing vectors in the list ##(u_1,u_2,...,u_n)## is...
36. ### Determining subspaces for all functions in a Vector space

Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...
37. ### Is R^n Euclidean Space a vector space too?

Dear Physics Forum personnel, I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n? Sincerely, PK
38. ### What is the difference between a vector and a vector space?

What is the difference between a vector and a vector space? I get that a vector is an object with both magnitude and direction, but am confused by the term "vector space". Does a vector space simply refer to a collection of vectors? Thanks!
39. ### What is vector space?

The thing is in my undergrad I haven't gone into a class that includes discussion about vector space, and the related stuffs, simply because they were not offered in the syllabus. As I have seen in some Quantum physics courses in some universities, they did talk about vector space in a certain...
40. ### Need help with proof of Vector Space (Ten Axioms)

Let S={x ∈ R; -π/2 < x < π/2 } and let V be the subset of R2 given by V=S^2={(x,y); -π/2 < x < π/2}, with vector addition ( (+) ). For each (for every) u ∈ V, For each (for every) v ∈ V with u=(x1 , y1) and v=(x2,y2) u+v = (arctan (tan(x1)+tan(x2)), arctan (tan(y1)+tan(y2)) ) Note: The...