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.
Hello!
1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors ##\vec a## and ##\vec b## : ##z=\vec a+ \vec bi##. I know that this complex vector can be visualized in 4D space. But I can't...
This is problem A.4 from Quantum Mechanics – by Griffiths & Schroeter.
I cannot make the Gram-Schmidt procedure work. I don't know whether I am just inept with complex vectors or I have made some wrong assumption.
The Gram-Schmidt procedure (modified, I think)
Suppose you start with a basis...
Summary:: summation of the components of a complex vector
Hi,
In my textbook I have
##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}##
##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}##
For ##\hat{e_p} = \hat{x}##...
Hi
Here is my attempt at a solution for problems 1) and 2) that can be found within the summary.
Problem 1)
a = 3-2i
b= -6-4i
c= 4+ 6i
d= -4+3i
Now, to calculate each vector modulus, I applied the following formula:
$$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$
where a = real part...
Homework Statement
let \epsilon_1 and \epsilon_2 be unit vectors in R3. Define two complex unit vectors as follows:
\epsilon_{\pm} = \frac{1}{\sqrt{2}}(\epsilon_1 \pm i \epsilon_2)
verify that epsilon plus minus constitutes a set of complex orthonormal unit vectors. That is, show that...
Homework Statement
Let a is a complex vector given by
a = 2π K - i ρ / α^2 ,
where ρ is a two dimensional position vector and K is the corresponding two dimensional vector in the Fourier space.
In order to find magnitude of this vector, i found that it is 4π^2 K^2 + ρ^2 / α^4 .
The logic...
So I was trying to learn how to find the angle between two complex 4-dimentional vectors. I came across this paper, http://arxiv.org/pdf/math/9904077.pdf which I found to be a little confusing and as a result not overly helpful. I was wondering if anyone could help at all?
Many thanks in...
Hi everyone,
I am interested how is polarized light absorbed by a molecule or an atom. Unfortunately, I come to a problem in the derivation where a complex vector in a real space appears. This is something I never seen before and I do not know how to interpret it. Therefore I would like to ask...
The way I understand it, they both have rectangular forms which are easy for addition/subtraction. Now I realize that the polar form of a complex vector can be simplified into an exponential, which is ideal for multiplication/division.
But this is what confuses me; vectors don't multiply/divide...
I've recently began a course on electromagnetism and have started dealing with complex vectors. I have a couple questions to ask:
Regarding the general concept of complex vectors, I am curious what these actually represent. Refer to attached equation. Am I correct to believe that this...
I was reading in my textbook that the scalar product of two complex vectors is also complex (I assuming this is true in general, but not in every case). However for the general definition (the inner product), each element of one of the vectors needs to be its complex conjugate. I learned this...
How is computed the cross product of complex vectors?
Let ##\mathbf{a}## and ##\mathbf{b}## be two vectors, each having complex components.
$$\mathbf{a} = a_x \mathbf{\hat{x}} + a_y \mathbf{\hat{y}} + a_z \mathbf{\hat{z}}$$
$$\mathbf{b} = b_x \mathbf{\hat{x}} + b_y \mathbf{\hat{y}} + b_z...
I have three (N x 1) complex vectors, a, b and c.
I know the following conditions:
(1) a and b are orthonormal (but length of c is unknown)
(2) c lies in the same 2D plane as a and b
(3) aHc = x (purely real, known)
(4) bHc = iy (purely imaginary, unknown)
where (.)H denotes...
I need some clarification on projections of complex vectors. If I have a nxm matrix of complex numbers V and a mx1 matrix s, and I want to find the projection of s onto any column of V. The formula to do this is
c = <V, s>/||V(j)||^2 where V(j) is the column of V to be used. My question is...
I was wondering, is the definition of the cross product the same for complex vectors? And if it is, then how is its geometric interpretation, that is
||\mathbf{a}\times\mathbf{b}||=||\mathbf{a}|| \; ||\mathbf{b}|| \sin\theta
Thanks.
Would you pls help me with the following vector product? I got no idea how the author derived the second equation from the first one. My derivation result is always including the imaginary unit i for the second term in the second equation on the right hand side. Specifically, how to verify that...
Homework Statement
Let z, w be complex vectors of C^n.
Prove ||w + z|| <= ||w|| + ||z||
(using the standard inner product for C^n)
(i.e. <w,z> = w*z', where * is the dot product and ' denotes the complex conjugate)
The Attempt at a Solution
Well, I found that
||w + z||
=...
What is the relationship btw the Hermitian inner product btw 2 complex vectors & angle btw them.
x,y are 2 complex vectors.
\theta angle btw them
what is the relation btw x^{H}y and cos(\theta)??
Any help will be good?
Statement:
<v(t)> = \frac{1}{T} \int^{T}_{0}v(t)dt = \frac{1}{T} \int^{T}_{0}V_0cos(\omega t + \phi)dt \equiv 0. (#1)Relevant Question:
If we suppose v(t) is a complex vector, is the second equality above still true?Reasoning:
If v(t) is a complex vector, then v(t) = cos(\omega t)\hat{x} +...
Homework Statement
Consider the unit vector, \hat{v}(t), expressed in instantaneous form:
\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} (#0)
The Vector will rotate counterclockwise in the x-y plane with angular velocity \omega.
Since both components are sinusoidally time varying...
Problem/Statement
The complex vector, \hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} is the unit vector \hat{v}(t) expressed in instantaneous form.
Question
What I am wondering is, why is there no imaginary component "j" in say the sin component for the equation above?
Can we...
Homework Statement
Can someone explain to me why complex vectors of the following form will rotate counterclockwise in the x-y plane [with a velocity of w],
\hat{v}(t) = cos(\omega t)\hat{x} + sin(\omega t)\hat{y} (#1)
And why the following equation, the unit vector rotates in the clockwise...
Hello,
I have recently started to study some Geometric Algebra.
I was wondering how should I interpret complex-vectors in \mathcal{C}^n in the framework of Geometric Algebra.
I understand already that a complex-scalar should be interpreted as an entity of the kind:
z = x + y (\textbf{e}_1...
Homework Statement
Find a basis for V=\mathbb{C}^1, where the field is the real numbers.
The Attempt at a Solution
I'd say \vec{e}_1=(1,0), \vec{e}_2=(i,0) is a basis, because it seems to me that \vec{u}=a+bi \in V can be written as...