What is Basis vectors: Definition and 76 Discussions
In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.
Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set.
A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
Assuming two forces applied on an object in two vector one horizontal to the object and the other vertical,
By vector addition the resulting vector would be in 45 degrees.
(0,1) + (1,0) = (1,1)
So the object acceleration direction would be as the following graph.
But if the force that...
The book I am studying from presents vector subspace as an infinite collection of vectors in a vector space with the properties of additive and multiplicative closure and basis vectors as a way to characterize/write a subspace compactly. All the vectors in the subspace can be written as linear...
There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...
A coordinate system with the coordinates s and t in R^2 is defined by the coordinate transformations: s = y/y_0 and t=y/y_0 - tan(x/x_0) , where x_0 and y_0 are constants.
a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system
is well defined. Express the...
a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)?
b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...
To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right?
$$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$
$$\hat{v}_z = \hat{v} \cdot \hat{z}$$
And do that for...
I'm reading a section on the derivative of a vector in a manifold. Quoting (the notation ##A^{\alpha}_{\beta'}## means ##\partial x^{\alpha}/\partial x^{\beta'}## - instead of using primed and unprimed variables, we use primed/unprimed indices to distinguish different bases):
Now this "we know...
On the way to Kruskal coordinates, Carroll introduces coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$$
{ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2
$$
##R_s=2GM## and we're using a ##-+++## signature...
I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors.
Quoting:
The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as:
$$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...
In the section 8-2 dealing with resolving the state vectors, we learn that
|\phi \rangle =\sum_i C_i | i \rangle
and the dual vector is defined as
\langle \chi | =\sum_j D^*_j \langle j |Then, the an inner product is defined as
\langle \chi | \phi \rangle =\sum_{ij} D^*_j C_i \langle j | i...
Hello all,
I was just introduced the Levi-Civita symbol and its utility in vector operations. The textbook I am following claims that, for basis vectors e_1, e_2, e_3 in an orthonormal coordinate system, the symbol can be used to represent the cross product as follows:
e_i \times e_j =...
Hello PF, here’s the setup: we have a geodesic congruence (not necessarily hypersurface orthogonal), and two sets of coordinates. One set, ##x^\alpha##, is just any arbitrary set of coordinates. The other set, ##(\tau,y^a)##, is defined such that ##\tau## labels each hypersurface (and...
You may be familiar with how you can express a vector field as a linear combination of basis vectors like so:
X = Xi∂i
Now, I know that normally, the basis vectors ∂i can be derived by taking the derivatives of the position vector for the coordinate system with respect to all the axes like...
Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc.
I'm going to define a tensor.
$$T^\mu_\nu \partial_\mu \otimes dx^\nu$$
Now I'd like to investigate how the tensor transforms...
Homework Statement
I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?
Standard basis vectors: e_1, e_2 or i,j
hi.
if I know how to convert coordinates from a system to cartesian system, then how can I find basevectors of that coordinatesystem?
Is it possible that basevectors are different in different points(with different coordinates)?
What is most general definition of basevectors? I tought it would...
Homework Statement
I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space.
Homework EquationsThe Attempt at a Solution
The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...
Homework Statement
I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates
Homework EquationsThe Attempt at a Solution
differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.
I read from this page https://properphysics.wordpress.com/2014/06/09/a-no-nonsense-introduction-to-special-relativity-part-6/
that the basis vectors are the canonical basis vectors in any coordinate system. This seems to be wrong, because if that was the case the metric would be the identity...
I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space.
1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ)
2. Tμνσρ = T(θμ,θν,eσ,eρ)
My attempt is as follows:
2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
In Schutz's A First Course in General Relativity (second edition, page 45, in the context of special relativity) he gives the scalar product of four basis vectors in a frame as follows:
$$\vec{e}_{0}\cdot\vec{e}_{0}=-1,$$...
Homework Statement
Let v(0) = [0.5 0.5 0.5 0.5]T, v(1) = [0.5 0.5 -0.5 -0.5]T, v(2) = [0.5 -0.5 0.5 -0.5]T, and z = [-0.5 0.5 0.5 1.5]T.
a) How many v(3) can we find to make {v(0), v(1), v(2), v(3)} a fully orthogonal basis?
b) What are z's coefficients of expansion αk in the basis found in...
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...
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...
Homework Statement
Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)##
a)Proof that this map is linear
b)Find some basis of the kernel
b)Find the dimension of the image
Homework Equations
##\mathbb{R}_{n}[x]##...
Is there a set of relationships for the wedge product of basis vectors as there are for the dot product and the cross product?
i.e. e1*e1 = 1
e1*e2 = 0
e1 x e2 = e3
What happens to a mechanical force's real original direction i.e. when we divide it into components of basis vectors, which in turn change as per problem at hand (like gravity components at inclined plane ), how we arrive at correct physics by taking two/three arbitrary directions of our choice...
I want to find basis for a+ax^2+bx^4 belong to p4.
I am getting the following result is it right?
=>a(1+x^2) + b(x^4)
=> basis ={1+x^2, x^4}
Is that right ? Please help me any help is appreciated.
I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922
If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...
Hi pf,
Having some trouble with basis vectors for expanding a given vector in 3-D space.
Any given vector in 3-D space can be given by a sum of component vectors in the form:
V = e1V1 + e2V2 + e3V3 (where e1, e2 and e3 are the same as i, j and k unit vectors). Equation 1.
I am happy with...
The wiki page https://en.wikipedia.org/wiki/Bell's theorem states the following which I agree with:
Suppose the two particles are perfectly anti-correlated—in the sense that whenever both measured in the same direction, one gets identically opposite outcomes, when both measured in opposite...
I'm trying to understand the maths of QM from Shankar's book - Principles of Quantum Mechanics: On page 21 of that book, there is a general derivation that if we have a relation:
|v'> = Ω|v>
Where Ω is a operator on |v> transfroming it into |v'>, then the matrix entries of the operator can be...
Hi, I'm doing a first course in GR and have just found out that
\eta_{ab} = g(\vec{e}_{a}, \vec{e}_{b}) = \vec{e}_{a} \cdot \vec{e}_{b}
where g is a tensor, here taking the basis vectors of the space as arguments. I haven't seen this written explicitly anywhere but does this mean that...
Hi. In GR , covariant differentiation is used because the basis vectors are not constant. But , what about in SR ? If the basis vectors are not Cartesian then they are not constant. Does covariant differentiation exist in SR ? And are for example spherical polar basis vectors which are not...
What is the dimensionality, N, of the Hilbert space (i.e., how many basis vectors does it need)?
To be honest I am entirely lost on this question. I've heard of Hilbert space being both finite and infinite so I'm not sure as to a solid answer for this question. Does the Hilbert space need 4...
Homework Statement
Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are...
Hello every one , in this pic i just printed ( Tensors_The Mathematics of Relativity Theory and Continuum Mechanics by Anadijiban Das ) here the author classifies the basis into 3 types 1- is the general basis (non-holomonic ) , 2- coordinate basis ( holomonic ),3- orthonormal basis ( non-...
Let me say from the beginning I'm not talking about the non-coordinate unit vectors for polar coordinates. I'm talking about basis vectors. Let me just ask it as boldly as possible: how does one use these basis vectors in order to describe a vector? I know they are different at every point, so...
In "A Student's Guide to Vectors and Tensors" by Daniel Fleisch, I read that the covariant metric tensor gij=ei°ei (I'm leaving out the → s above the e's) where ei and ei are coordinate basis vectors and ° denotes the inner product, and similarly for the contravariant metric tensor using dual...
I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain:
We have our usual polar coordinates relation to Cartesian:
x = r cosθ ; y = r sinθ
if I define \hat{e_{r}}, \hat{e_{\vartheta}} as the polar basis vectors, then they should be...
Hello all, I don't have a question on homework specifically, but I need clarification on something I'm reading in the textbook.
I will be starting an abstract algebra class in the spring and it's been quite a few years since I've had linear algebra, so I'll be reviewing that material before the...
I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that:
"It is straightforward to show that the coordinate and dual basis vectors
themselves are related...
"ea = gabeb ..."
I have...
Homework Statement
A vector is a geometrical object which doesn't depend on the basis we use to represent it, only its components will change. We can express this by \vec{A}=ƩA_i \hat{ε_i} = Ʃ\tilde{A_i} \vec{ε_i}, where it has been emphasized that the basis ε is not necessarily orthonormal...
Firstly; is there a difference between the "regular" polar coordinates that use \theta and r to describe a point (the one where the point (\sqrt{2}, \frac{\pi}{4}) equals (1, 1) in rectangular coordinates) and the ones that use the orthonormal basis vectors \hat{e}_r and...
Homework Statement
The problem is Exercise 2 in the picture
http://postimage.org/image/3ou3x1sh7/
Homework Equations
The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors?
The Attempt at a Solution
I have no idea where...
Greetings,
I have just started studying manifolds, and have come across the idea that the basis vectors can be expressed as:
e\mu = \partial/\partialx\mu.
I tried to convince myself of this in 2D Cartesian coordinates using a pretty non-rigorous derivation (the idea being to get a...