Linear independence

  1. Any one plz tell me about the term linear independence?and when we say that the function is linear independent
  2. jcsd
  3. mathman

    mathman 6,748
    Science Advisor
    Gold Member

    You need to supply context. Usually linear independence refers to a set of vectors or a set of functions. These things are called linearly dependent if some linear combination = 0. If not then they are linearly independent.
  4. [A][/A]=[c][/1]x+[c][/2]y[j][/j]+[c][/3]z[k][/k]
    when we say it is linearly independent
    and also my friend argue with me that orthonormality implies linear independence but i was not satisfied plz help :smile:
  5. mathman

    mathman 6,748
    Science Advisor
    Gold Member

    I don't understand your first line.

    However your friend is correct, if two vectors are orthogonal (unless one of the is 0) they are linearly independent. Note that the converse is not true.
  6. Fredrik

    Fredrik 10,634
    Staff Emeritus
    Science Advisor
    Gold Member

    A set E is said to be linearly independent if for all finite subsets ##\{x_1,\dots,x_n\}\subset E## and all ##a_1,\dots,a_n\in\mathbb C##,
    $$\sum_{k=1}^n a_k x_k=0\quad \Rightarrow\quad a_1=\dots=a_n=0.$$
    Suppose that E is orthonormal. Let ##\{e_k\}_{k=1}^n## be an arbitrary finite subset of E, and suppose that ##\sum_{k=1}^n a_k e_k=0##. Then for all ##i\in\{1,\dots,n\}##,
    $$0=\langle e_i,0\rangle=\langle e_i,\sum_k a_k e_k\rangle=\sum_k a_k\langle e_i,e_k\rangle=\sum_k a_k\delta_{ik}=a_i.$$
  7. Geometrically, linear independence means each vector contributes something to the span of the vectors.

    So, if you have one vector, it spans a 1-dimensional subspace. If you add another vector to the set, you get a linearly dependent set if the span doesn't get any bigger. So, two vectors are linearly dependent if one is a multiple of the other. So, if you added a vector that was pointing in a different direction (and not the opposite direction), together they span a 2-dimensional subspace. In that case, they are said to be linearly independent.

    So, in general, if you have n vectors, they are linearly independent if throwing one of them out makes them span a smaller subspace. If you can throw one out without changing the span, they are linearly dependent.

    This picture is less accurate, but still helpful, for more general vector spaces, in which the "vectors" aren't exactly "arrows" pointing in space anymore.

    Functions are really a kind of vector because vectors are things that you can add together and multiply by scalars. So, it's the same idea for functions.
  8. Thank to all of you
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