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Linear Transformation: Find the matrix

  1. Jan 6, 2016 #1
    1. The problem statement, all variables and given/known data
    Let A(l) =
    [ 1 1 1 ]
    [ 1 -1 2]
    be the matrix associated to a linear transformation l:R3 to R2 with respect to the standard basis of R3 and R2. Find the matrix associated to the given transformation with respect to hte bases B,C, where
    B = {(1,0,0) (0,1,0) , (0,1,1) }
    C = {(1,1) , (1,-1)}
    2. Relevant equations
    T(x) = Ax
    L(x,y,z) = (ax+by+cz, dx+ey+fz)
    3. The attempt at a solution
    I don't full understand what this question is asking. The matrix A(l) is the matrix A in the equation, T(x) = Ax, no? Then I'm supposed to matrix multiply A by the standard basis in R3 and R2? ie:

    [ 1 0 0 ]
    [ 0 1 0] * A
    [ 0 0 1]

    I don't think this is correct because I can't do this for R2 because that's a 2x2 matrix and the matrix A is a 2x3 matrix.
     
  2. jcsd
  3. Jan 6, 2016 #2

    Mark44

    Staff: Mentor

    The matrix at the top is the matrix of L in terms of the standard bases for ##\mathbb{R}^3## and ##\mathbb{R}^2##, which are, respectively {<1, 0, 0>, <0, 1, 0>, <0, 0, 1>} and {<1, 0>, <0, 1>}.

    What you're asked to do is to find the matrix representation in terms of the B and C bases.
     
  4. Jan 6, 2016 #3
    T(x) = A(x)

    [ 1 1 1 ] = Matrix I need to find * Standard basis in R3
    [ 1 -1 2]

    [ 1 1 1 ] = Matrix I need to find * Standard basis in R2
    [ 1 -1 2]

    ??
     
  5. Jan 6, 2016 #4

    haruspex

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    It will be easier to change one base at a time. Let's start with replacing the R3 basis.
    For clarity, I'll use upper case for representations of vectors and lowercase for the abstract versions.
    Suppose you had a matrix M that converts a vector v written in terms of the new basis (V' say) to the same vector written in terms of the old basis (V). If, in the old basis, Y=AV, how can you write that using V' instead of V?
     
  6. Jan 6, 2016 #5
    Sorry, I don't fully understand your question. The notation is a little confusing.

    V is the standard basis, but I should matrix multiply them one at a time, like (1,0,0) and then (0,1,0) ...
    A is the matrix that I have at the minute?
     
  7. Jan 6, 2016 #6

    Mark44

    Staff: Mentor

    Doesn't your textbook have an example like this? If you don't understand this process, that should be the first place you look, not here.
     
  8. Jan 6, 2016 #7
    Matrix I'm looking for (2x3) * Standard basis in R3 = A(l)

    Standard basis in R2* Matrix I'm looking for (2x3) = A(l)
     
  9. Jan 6, 2016 #8
    Let's not talk about the "textbook"...

    I understand linear transformations. But it's when I'm asking a question it just throws me. They could be written in another language and they'd make as much sense.
     
  10. Jan 6, 2016 #9

    Ray Vickson

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    No, if you really what you wrote then that means that you do NOT understand linear transformations.

    You say "let's not talk about the textbook" Why not?

    Anyway, if the textbook is so bad you can always find good alternatives on-line. Google "linear transformations" or "linear algebra" or similar titles to find numerous sources. I believe there are even free textbooks in pdf form--it is all there, you only need to look.
     
  11. Jan 6, 2016 #10
    My textbook is absolutely horrible...
    I don't understand what the question is asking, it's mapping a vector from R3 to R2. I know with linear transformations that matrix multiplication w/ vectors is always a linear transformation. I'm just a bit confused because the equation I've been working with is T(x) = A(x), where x is a vector and A is a matrix, but this question has the matrix as A(l)....

    edit: and I've looked at other online sources, but I can't seem to find anything / anyone that can explain questions like this in very very simple terms. From there I can build my understanding and then replace the words / meaning I have with a more mathematically formal definition.
     
  12. Jan 6, 2016 #11

    Mark44

    Staff: Mentor

    Whether it's actually horrible or not, your textbook should have some examples of the change of basis for a linear transformation. Every linear algebra book I've ever seen has a section on the change of basis. Again, your first resource should be your textbook, regardless of your opinion of it.
    It really should be T(x) = Ax. On the left side, x is the argument of the transformation T, just like regular function notation. On the right side it's the matrix A multiplying the vector x.
    That notation means "the matrix representation of the transformation l (letter ell)" -- that's all it means.
     
  13. Jan 6, 2016 #12

    Ray Vickson

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    This is one reason I have (in several responses to your past posts) recommended that you avoid using matrices until you really grasp what is going on. Once you understand what linear transformations are about then---and only then--- should you start to talk about matrices, which are really just bookkeeping devices to shorten notation and make things easier to write (at least at the beginning stages of study). When you reach the higher levels in these topics then matrices play a much more important role and become crucial to several types of analyses. However, right down at the beginning stage you can get along just fine without ever using a matrix, and maybe even enhance your understanding that way.

    Naturally, you do have arrays of constants that, for example, express one set of basis vectors in terms of another, or an array of coefficients that tell you how vectors transform for a given linear transformation, but as I said, matrices (at this level) are essentially just booking rules that keep track of these coefficients. The thing to understand first is how to get these cited coefficients in a particular case; later you can form them up into a matrix if you really feel the need to do so.
     
  14. Jan 6, 2016 #13

    fresh_42

    Staff: Mentor

    Would a real world example help you? The numbers in question, the coordinates, whether they are those of the vectors or those of the linear transformations have to be defined in a framework. The starting point has been the standard basis, i.e. the 3 orthogonal spread fingers of your right hand. Every point in the room can then be described on how many finger length it takes in every direction to reach it. But if you change the angles between your fingers or you take all of a sudden your left hand then all the numbers will describe a different point. They have to be adjusted to the new framework. That's what the question is about: What happens if you change the system in which the coordinates are measured.
     
  15. Jan 6, 2016 #14

    haruspex

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    Not sure where your difficulty starts, so I'll start at the beginning.
    A vector in a vector space is an abstract concept. A representation of it as a list of scalar values is only meaningful in the context of a chosen basis, an independent set of vectors within the space. Unfortunately, as far as I am aware, there is no separate term for such a representation, it is also called a vector. So I adopted lowercase for the abstract form and uppercase for a representation. Thus my V and V' represent the same vector v, but using the two different bases. Thus, if the bases are the (abstract) vector lists (e1,..,en) and (e'1,..,e'n) then we have the abstract vector equation ##v=\Sigma V_ie_i=\Sigma V'_ie'_i##.
    Likewise, a linear transformation is an abstract function from one vector space to another (or to itself). Given a basis for each vector space, it can be represented as a matrix.
    You have a transformation, l (lowercase L), represented by the matrix A given the basis (e1,..,en) in the domain space and some basis in the range space. Since that comes out looking like I in the font I'm using, I'll call it a instead.

    (You wrote something like T(x)=Ax, but that is not right strictly. Using my case-sensitive notation, it becomes a(x)=AX, but that is still wrong because the left hand side is an abstract vector and the right hand side is not.)

    Now, let's leave the range space basis fixed for the moment.
    You have a matrix A, and AX produces the representation of a(x), where X is the representation of x in the original basis. Now we are given, instead, X' as the representation of x in a new basis. In order to use the matrix A, we must first discover the representation of x in the old basis. So you need to figure out how to convert an X' to an X. I.e. you wish to map one basis to the other. This will be a matter of finding a matrix to do it. The tricky part is getting the mapping the right way round!

    Edit: I see the font came out ok after all. In what I type, l and I look the same, but once posted they're different.
     
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