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Confused about vectors and transformations (linear)

  1. Feb 13, 2015 #1
    when we are talking about a linear transformation the argument of the function is a coordinate vector...is this true?
    another question...when i see a column vector...these are the coordinates of the vector with respect of a basis...is this true? for example if i see....

    [itex]
    (({{1},{3}}))^T
    [/itex]

    with respect to a basis [itex] {a1,a2} [/itex]...that "column symbol" are the coordinates of this vector?:
    [itex] v=a1*1+a2*3 [/itex]
    ??
     
    Last edited: Feb 13, 2015
  2. jcsd
  3. Feb 13, 2015 #2

    Stephen Tashi

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    Science Advisor

    Abstractly, one can talk about a linear transformation as a function whose argument is a single symbol, such as [itex] T(X) [/itex] where [itex] X [/itex] is a vector. Linearity requirements such as [itex] T(\alpha X + Y) = \alpha T(X) + T(Y) [/itex] can be stated without reference to particular coordinates. (For example, in 2-D polar coordinates, the above requirement can be implemented by a coordinate operations that are different from an implementation in cartesian coordinate operations.)

    I'd say yes.

    When a vector is expressed in terms of a linear combination of a particular (finite) set of basis vectors then a customary way to represent a vector [itex] X [/itex] is as a column vector whose entries are the scalars in the linear combination. (So it is fair to call that representation a "coordinate representation", although it does not necessarily refer to coordinates Euclidean space.) Then a linear transformation [itex] T [/itex] amounts to left multiplying the column vector by a matrix of constants.
     
  4. Feb 13, 2015 #3
    ok, thanks a lot!
     
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