Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear Transformations and Bases

  1. Apr 28, 2012 #1
    I need some help or at least some assurance that my thinking on linear transformations and their matrix representations is correct.

    I assume when we specify a linear transformation eg F(x,y, z) = (3x +y, y+z, 2x-3z) for example, that this is specified by its action on the variables and is not with respect to any basis.

    However when we specify the matrix of a linear transformation T: V --> W that this is with respect to a basis in V and a basis in W

    Of course if we have a linear transformation S: V -->V it could be that the two bases are the same.

    If no basis is mentioned regarding the matrix of a linear transformation, then I am assuming the standard bases are assumed.

    Can someone either confirm I am correct in my thinking or point out the errors in my thinking?

  2. jcsd
  3. Apr 28, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What is the standard basis of a generic vector space V?
  4. Apr 28, 2012 #3
    Yes, good point ... I guess I should have specified Euclidean space for that assumption to make sense ...

    Are my other assumptions/interpretations OK?
  5. Apr 28, 2012 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Everything is correct, but I think you have a bit of a hangup on how linear transformations on vector spaces are described

    The key here is that in your first example you specified a linear transformation on R3 without defining it as a matrix multiplication, so no basis is required and no matrix is ever constructed. If you wanted to define F as a matrix multiplication you would need to specify a basis of R3 - on Euclidean space this step is often omitted because everyone assumes your basis is the standard basis (1,0,0),(0,1,0),(0,0,1) (and trying to specify F as matrix multiplication with respect to a different basis is literally just extra work).

    If we want to specify a linear transformation V--> W as a matrix multiplication we need to pick bases of V and W to identify them with Euclidean space. But we can have linear transformations that are not represented as matrix multiplications. For example let V be the set of all polynomials of degree <= 3 (this is a 4 dimensional vector space over R) and let W be R. Then consider
    [tex] I:V\to \mathbb{R},\ I(p(x)) = \int_0^1 p(x) dx [/tex]
    I is a linear transformation and I never specified a basis for V in order to tell you the function because I didn't tell you what I was as a matrix multiplication
  6. Apr 28, 2012 #5
    Thanks so much for that post - most helpful

    I suppose the essential thing needed to be able to derive a matrix of a linear transformation is that the vector spaces involved need to be over a field F where F = R or C.

    Is that correct?

    Missed your example due to some latex error or other - pity - would have like to have viewed your example

    Thanks again!

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook