What is Vector space: Definition and 538 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. N

    B Understanding Bases of a Vector Space

    In the book I'm reading, Before Machine Learning, by Jorge Brasil, I'm on the section that introduces bases for vector spaces. The author gives the example of a vector space with two vectors ##\vec i## and ##\vec j## forming the basis where ##\vec i = (1,0)## and ##\vec j = (0,1)## He then says...
  2. cianfa72

    I Inner product vs dot/scalar product

    Hi, from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things. Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
  3. Z

    Do these two statements imply an underlying induction proof?

    Here is one proof $$\forall u\in U\implies Tu\in U\subset V\implies T^2u\in U\implies \forall m\in\mathbb{N}, T^m\in U\tag{1}$$ Is the statement above actually a proof that ##\forall m\in\mathbb{N}, T^m\in U## or is it just shorthand for "this can be proved by induction"? In other words, for...
  4. M

    Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

    Determine whether the following subsets U of M4x4is a subspace of the vector space V of all M4x4 matrices, with the standard operations of matrix addition and scalar multiplication. If is not a subspace provide an example to demonstrate a property that U does not possess. a. The set U of all 4x4...
  5. C

    Does each norm on vector space become discontinuous when restricted to S^1?

    Dear Everybody, I am having trouble with last part of this question. I believe the answer is no. But I have to proof the general case. Here is my work for the problem: Suppose that we have two distinct norms on the same vector space ##X## over complex numbers. Then there exists no ##K## in...
  6. PhysicsRock

    How to Show Linearity of a Function?

    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...
  7. D

    I Prove $$T_{p}M$$ is a vector space with the axioms

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  8. J

    Linear operator in 2x2 complex vector space

    Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
  9. H

    Vector space of functions defined by a condition

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  10. P

    Prove that ##S## is a subspace of ##V##

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  11. H

    Prove that the inner product converges

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  12. Gere

    I Are density matrices part of a real vector space?

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  13. Leo Liu

    Vector space and 3D flow field

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  14. Santiago24

    I Two ways to define operations in a vector space

    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?
  15. H

    A Wick's rotation on a complex vector space

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  16. appletree23

    Help with linear algebra: vectorspace and subspace

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  17. S

    Vector space of linear maps

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  18. LCSphysicist

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  19. F

    Vector space of functions from finite set to real numbers

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  20. LCSphysicist

    Prove a theorem about a vector space and convex sets

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  21. K

    Linear algebra inner products, self adjoint operator,unitary operation

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  22. T

    MHB Proving Zp is a Vector Space for Prime p

    How can I prove that Zp is a vector space if and only if p is prime
  23. F

    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 ...
  24. penroseandpaper

    I Showing a set is a basis for a vector space

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  25. P

    A Double Dual of Vector Space: Is V** Always Same as V?

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  26. T

    I Vector space for solutions of differential equations

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  27. B

    I Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2

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  28. V

    Determining whether a set is a vector space

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  29. G

    B Understanding Dual Space: Mapping Vector Space to Real Numbers

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  30. M

    Vector space - polynomials vs. functions

    As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...
  31. M

    Why do we try to find if a subset is a subspace of a vector space?

    I am assuming the set ##V## will have elements like the ones shown below. ## v_{1} = (200, 700, 2) ## ## v_{2} = (250, 800, 3) ## ... 1. What will be the vector space in this situation? 2. Would a subspace mean a subset of V with three or more bathrooms?
  32. S

    I Showing direct sum of subspaces equals vector space

    If one shows that ##U\cap V=\{\textbf{0}\}##, which is easily shown, would that also imply ##\mathbf{R}^3=U \bigoplus V##? Or does one need to show that ##\mathbf{R}^3=U+V##? If yes, how? By defining say ##x_1'=x_1+t,x_2'=x_2+t,x_3'=x_3+2t## and hence any ##\textbf{x}=(x_1',x_2',x_3') \in...
  33. P

    I "Adding" a Vector Space and its Dual

    Given that the Set of 1-Forms is a Vector Space distinct from, but complimentary to, the Linear Vector Space of Vectors. And given that there is an Isomorphism between the linear space of vectors and the dual vector space of 1-forms, does it make mathematical sense to combine a vector space and...
  34. Eclair_de_XII

    If V is a vector space why is T^2(V) = T(V) iff ker(T^2) = ker(T)?

    --##ker(T^2)=ker(T)## if ##T(V)=T^2(V)##-- Suppose that ##T^2(V)=T(V)##. So ##T:T(V)\mapsto T^2(V)=T(V)##. Hence, ##T## is one-to-one and so ##ker(T)=\{0\}##. Suppose that ##T^2(w)=0## for some ##w\in ker(T^2)##. Then ##T^2(w)=T(T(w))=0## which implies that ##T(w)\in ker(T)## and so ##T(w)=0##...
  35. Math Amateur

    MHB Open Balls in a Normed Vector Space .... Carothers, Exercise 32

    I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms ... ... I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows: I have not been able to make much progress ... We have ...B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}...
  36. M

    A Vector space (no topology) basis

    The standard definition of the basis for a vector space is that all the vectors can be defined as finite linear combinations of basis elements. Consider the vector space consisting of all sequences of field elements. Basis vectors could be defined as vectors which are zero except for one term in...
  37. karush

    MHB 10.2 Determine if the set of vectors form a vector space

    Determine if the set of vectors $\begin{bmatrix} x\\y\\5 \end{bmatrix}\in \Bbb{R}^3$ form a vector space ok if I follow the book example I think this is what is done $\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5...
  38. U

    Vector space has dimension less than d

    Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...
  39. Math Amateur

    I Vector Space of Alternating Multilinear Functions ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...
  40. M

    MHB Is f in the vector space of cubic spline functions?

    Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...
  41. karush

    MHB Set of vectors form a vector space

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  42. karush

    MHB Determine if vector space.

    On the set of vectors $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}\in \Bbb{R}^2 $ with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by $$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} \oplus \begin{bmatrix} x_2 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \\...
  43. GlassBones

    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...
  44. V

    MHB Can Vector Space $(V,O_1,O_2)$ Represent 2 Graphs?

    Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...
  45. M

    MHB The axioms of a vector space are satisfied

    Hey! :o We consider the $\mathbb{F}_2$-vector space $(2^M, +, \cap)$, where $M$ is non-empty set and $+ : 2^M\times 2^M \rightarrow 2^M: (X,Y)\mapsto (X\cup Y)\setminus (X\cap Y)$. I want to show that $(2^M, +, \cap )$ for $\mathbb{K}=\{\emptyset , M\}$ satisfies the axioms of a vector space...
  46. M

    MHB Vector space - Prove or disprove

    Hey! :o Let $1\leq n\in \mathbb{N}$ and let $U_1, U_2$ be subspaces of the $\mathbb{R}$-vector space $\mathbb{R}^n$. I want to prove or disprove the following: The set $\{f\in \mathbb{R}^{\mathbb{R}} \mid \exists x\in \mathbb{R} : f(x)=0_{\mathbb{R}}\}$ is a subspace of...
  47. hilbert2

    A Linearly independent function sets

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  48. karush

    MHB 307.8.1 Suppose Y_1 and Y_2 form a basis for a 2-dimensional vector space V

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  49. Pushoam

    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...
  50. Pushoam

    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...
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