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.

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  1. matqkks

    Unlock the Power of Basis Vectors: Impactful Examples

    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...
  2. matqkks

    MHB How to introduce basis vectors

    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.
  3. T

    If V is a 3-dimensional Lie algebra with basis vectors E,F,G

    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...
  4. S

    Undergraduate Mechanics (Problem with force expressed as basis vectors)

    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...
  5. V

    Capturing n basis vectors by single one

    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...
  6. B

    Basis vectors and abstract index notation

    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...
  7. S

    Difficult theoretical problem on basis vectors

    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!
  8. Matterwave

    Pondering basis vectors and one forms

    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...
  9. DryRun

    Is a set of orthogonal basis vectors for a subspace unique?

    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...
  10. A

    Regarding normalization of the eigen basis vectors

    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...
  11. R

    Partial derivatives as basis vectors?

    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...
  12. K

    Basis vectors under a Lorentz transformation

    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...
  13. J

    Comparing Basis Vectors in Linear Spaces: X and Y

    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...
  14. snoopies622

    Basis Vector Length: Unit Length & Mistakes to Avoid

    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?
  15. E

    Basis vectors and ortho solution spaces

    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)]...
  16. snoopies622

    Understanding Basis Vectors for Tensor Differentiation

    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 =...
  17. F

    Does Every Coordinate System Admit Local Othornormal Basis Vectors

    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...
  18. P

    Orthogonormal and Basis Vectors

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

    Standard basis vectors of C^n?

    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.
  20. E

    Transformations of Basis Vectors on Manifolds

    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...
  21. R

    Uncertainty and basis vectors of relativity

    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...
  22. W

    How do we express basis vectors for a manifold in terms of partial derivatives?

    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...
  23. L

    Programs for Converting Basis Vectors

    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 :)
  24. F

    Dot product of basis vectors in orthogonal coordinate systems

    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...
  25. M

    Can basis covectors be defined without a metric?

    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...
  26. C

    Linear independence of basis vectors

    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...
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