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.
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...
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...
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...
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...
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...
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...
Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms:
Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space
$$u + v \in V \tag{Closure under addition}$$
$$u + v = v +...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
##f : [0,2] \to R##. ##f## is continuous and is defined as follows:
$$
f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$
$$
f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$
##V = \text{space of all such f}##
What would the basis for V? Well, for ##x \in [0,1]## the basis for ##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...
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##...
Is the following a correct demonstration that quantum mechanics can be done in a real vector space?
If you simply stack the entries of density matrices into a column vector, then the expression ##\textrm{Tr}(AB^\dagger)## is the same as the dot product in a complex vector space (Frobenius inner...
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?
I read this in the wiki article about Wick rotation:
Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no...
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...
Solution
1. Based on my analysis, elements of ##V## is a map from the set of numbers ##\{1, 2, ..., n\}## to some say, real number (assuming ##F = \mathbb{R}##), so that an example element of ##F## is ##x(1)##. An example element of the vector space ##F^n## is ##(x_1, x_2, ..., x_n)##.
From...
Be ##T_{1}, T_{2}## upper and lower matrix, respectivelly. Show that we haven't matrix ##M(NxN)## such that ##M(NxN) = T_{1}\bigoplus T_{2}##
I am not sure if i get what the statement is talking about, can't we call ##T_{1},T_{2} = 0##? Where 0 is the matrix (NxN) with zeros on all its entries...
Summary:: Problem interpreting a vector space of functions f such that f: S={1} -> R
Hello,
Another question related to Jim Hefferon' Linear Algebra free book. Before explaining what I don't understand, here is the problem :
I have trouble understanding how the dimension of resulting space...
Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1
I tried to suppose...
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!
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 ...
If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B?
I have calculated another basis of B, and found I can use linear combinations...
Hi
I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**.
What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...
Good Morning
Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you.
In it, someone wrote:
" It is a theorem in mathematics that the set of all functions that are solutions of a...
This exercise is located in the vector space chapter of my book that's why I am posting it here.
Recently started with this kind of exercise, proof like exercises and I am a little bit lost
Proof that given a, b, c real numbers, the set X = {(x, y) E R^2; ax + by <= c} ´is convex at R^2
the...
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...
I understand that the Dual Space is composed of elements that linearly map the elements of the Vector Space onto Real numbers
If my preamble shows that I have understood correctly the basic premise, I have one or two questions that I am trying to work through.
So:
1: Is there a one to one...
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...
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?
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...
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...
--##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##...
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 \}...
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...
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...
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...
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...
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 |$$ is in $S_{X,3}$...
this is what is given
so by addition
$$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix}
\oplus
\begin{bmatrix} x_2\\y_2\\5z_2
\end{bmatrix}
=
\begin{bmatrix}
x_1+x_2\\y_1+y_2\\5z_1+5z_2
\end{bmatrix}
=
\begin{bmatrix}
X\\Y\\10Z
\end{bmatrix}$$
uhmmmm really?
On the set of vectors
$\begin{bmatrix}
x_1 \\ y_1
\end{bmatrix}\in \Bbb{R}^2 $
with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by
$$\begin{bmatrix}
x_1 \\ y_1
\end{bmatrix} \oplus
\begin{bmatrix}
x_2 \\ y_2
\end{bmatrix}
=
\begin{bmatrix}
x_1 + x_2 \\...
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...
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...
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...
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...
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...
nmh{796}
$\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$
$\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$
$$Y_1=\begin{bmatrix}a\\b\end{bmatrix}
\textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$
$\textit{ then }$...
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...
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...