Basis vectors Definition and 76 Threads

I have normally introduced basis vectors by just stating independent vectors that span the space. This is perhaps not very inspirational. What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. Maybe a good...

What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. I have normally introduced it by just stating independent vectors that span the space.

If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...

Homework Statement The problem along with its solution is attached as Problem 1-2.jpg. Homework Equations Norm of a vector. The Attempt at a Solution Starting from the final answer of the solution, sqrt((-0.625)^2 + (0.333)^2) == 0.708176532 != 1. Did the book do something wrong? I ask...

Normally, you need how system transforms n basis vectors to say how it transforms arbitrary vector. For instance, when your signal is presented in Fourier basis, you need to know how system responds to every sine. But, I have noted that it is not true for the simplest standard basis. You just...

First of all, I'd like to say hi to all the peole here on the forum! Now to my question: When reading some general relativity articles, I came upon this strange notation: T^{a}_{b} = C(dt)^{a}(∂_{t})_{b} + D(∂_{t})^{a}(dt)_{b}. Can someone please explain to me what this means? Clearly...

How the hell do you prove that the components of a vector w.r.t. a given basis are unique? I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

So, I've been thinking about this for a while...and I can't seem to resolve it in my head. In this thread I will use a tilde when referring to one forms and a vector sign when referring to vectors and boldface for tensors. It seems to me that if we require the basis vectors and one forms to obey...

Homework Statement Is a set of orthogonal basis vectors for a subspace unique? The attempt at a solution I don't know what this means. Can someone please explain? I managed to find the orthogonal basis vectors and afterwards determining the orthonormal basis vectors, but I'm not sure what the...

For a continuous eigen-basis the basis vectors are not normalizable to unity length. They can be normalized only upto a delta function. At the same time for discrete eigen basis the basis vectors are normalizable to unity length. What about the systems with both discrete as well as continuous...

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Hello, I am new to the forums and I hope this fundamental topic has not been previously treated, as these forums don't seem to have a search function. I am studying general relativity using S. Carroll's book (Geometry and Spacetime) and I am having a fundamental problem with basis vectors under...

Hi everyone, I am working on the following problem. Suppose the set of vectors X1,..,Xk is a basis for linear space V1. Suppose the set of vectors Y1,..,Yk is also a basis for linear space V1. Clearly the linear space spanned by the Xs equals the linear space spanned by the Ys. Set X=[X1: X2...

For an ordinary vector V, the square of its length is V \cdot V = V^a V_a. For basis vectors, e^a \cdot e_b = \delta ^a _b so e^a \cdot e_a = 1. Since 1^2 = 1, this implies that every basis vector is of unit length. What is my mistake?

Hello, I've got two homogenous equations: 3x + 2y + z - u = 0 and 2x + y + z +5u = 0. I'm trying to find a basis for these solutions. The solution vector x [x, y, z, u] is a solution if and only if it is orthogonal to the row vectors, in this case a and b ([3, 2, 1, -1], [2, 1, 1, 5)]...

This is still rather new to me so please pardon my ignorance. My introduction to tensor differentiation involved only manifolds that were embedded in higher-dimensional Euclidean spaces. To describe them, I was instructed to find basis vectors using partial derivatives, as in e_\theta =...

I wonder if there are coordinate systems that gobally curve and twist and turn and curl, that do NOT admit local orthonormal basis. I know that the Gram-Schmidt procedure converts ANY set of linear independent vectors into an orthnormal set that can be used as local basis vectors. And I assume...

Would all bases be sets of orthogonal (but not necessarily orthonormal) vectors?

I take it that the standard basis vectors of C^n is the same as the standard basis vectors of R^n? It would seem so as scalars in C^n are complex numbers.

Homework Statement I am trying to show that \vec{e'}_a = \frac{\partial x^b}{\partial x'^a} \vec{e}_b where the e's are bases on a manifold and the primes mean a change of coordinates I can get that \frac{\partial x^a}{ \partial x'^b} dx'^b \vec{e}_a = dx'^a \vec{e'}_a from the invariance...

Do the basis vectors of Einstein's General Theory of Relativity have the same point of origin, given that the uncertainty principle says that we can't know exactly the position of something? And if we say the basis vectors do have the same point of origin isn't this the same as introducing bias...

In introductory mechanics courses we derive the equations of motion in curvilinear coordinates, especially the m (d^2/dt^2)x, by expressing the coordinate basis vectors in terms of their cartesian counterparts, and then differentating them with respect to time. For example, in 2D polar...

I wasn't quite sure where to post this but... Does anyone know of a good program that will convert vectors from cartesian to spherical or cylinrical. I am (tediously) doing it by hand and would like to check my work. Thanks :)

I'm doing a series of questions right now that is basically dealing with the dot and cross products of the basis vectors for cartesian, cylindrical, and spherical coordinate systems. I am stuck on \hat R \cdot \hat r right now. I'll try to explain my work, and the problem I am running into...

I often think I have fully understood this, then some question comes up in my mind, and I get confused again (which implies I never understood it in the first place). We have a co-ordinate basis for vectors {\partial_\mu}. I can think of two ways to get a corresponding basis for covectors. 1...

How do I prove the linear independence of the standard basis vectors? My book is helpful by giving the definition of linear independence and a couple examples, but never once shows how to prove that they are linearly independent. I know that the list of standard basis vectors is linearly...