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Linear Transformations notation

  1. Apr 11, 2012 #1
    Hi Pf,

    Here is a question regard a test review that we have. I am not looking for the answer but rather a clarification about the notation.

    1. What does the following mean? T1: [itex]\Re[/itex]2 [itex]\rightarrow[/itex] [itex]\Re[/itex]2 by x [itex]\rightarrow[/itex] Ax?

    2. What does it mean to go [itex]\Re[/itex]2 [itex]\rightarrow[/itex] [itex]\Re[/itex]2

    Thanks.
     
  2. jcsd
  3. Apr 11, 2012 #2

    chiro

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    Hey DmytriE.

    Basically the LHS of the arrow is your starting space and the RHS is your target space. In other words we are starting in R^2 (2D vector with real numbers in each element) and we are going to a 2D vector.

    In terms x -> Ax, this means that we start with a vector x and then we apply the operator A to the vector x by calculating Ax using normal matrix multiplication to get a new vector (still in R^2) called x' where x' = Ax.
     
  4. Apr 12, 2012 #3
    Thanks Chiro!
     
  5. Apr 12, 2012 #4

    Fredrik

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    The technical terms for "starting space" and "target space" are "domain" and "codomain". The notation ##\mathbb R## (\mathbb R) is more common than ##\Re## (\Re).

    It strikes me as a bit odd to write "##T_1:\mathbb R^2\to\mathbb R^2## by ##x\to Ax##". The notation ##T_1:\mathbb R^2\to\mathbb R^2## tells us that ##T_1## is a function with domain ##\mathbb R^2## and codomain ##\mathbb R^2##. The notation ##x\to Ax## should mean "the function that takes x to Ax". Wouldn't you denote that function by A? Hm, I guess that the most likely explanation is that A is a matrix, and the person who wrote this would like to emphasize that the linear operator T1 that corresponds to the matrix A isn't the same thing as A (even though T1 acting on x gives us the same result as A times x).

    By the way, I prefer to use the \mapsto arrow in the second notation, i.e. I would write ##x\mapsto Ax## instead of ##x\to Ax##. Some people prefer to never use the mapsto arrow. That's OK too.
     
  6. Apr 12, 2012 #5
    I was trying to us the \mapsto arrow but when I clicked on it it gave me the bidirectional arrow and I did not know it's proper name so I just selected the normal arrow.

    Here is another thing that confused me and my professor explained but I guess I was unable to grasp it. What is the difference between onto and one-to-one? What quantities would you look at to determine if it is one-to-one or onto (i.e. Null(A) or Col(A))? An example would help a lot.
     
  7. Apr 12, 2012 #6

    Fredrik

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    A function f:X→Y is said to be injective (or one-to-one) if for all x,y in X, f(x)=f(y) implies x=y.

    A function f:X→Y is said to be surjective (or onto) if for all y in Y, there's an x in X such that f(x)=y.

    You can find examples in the wikipedia articles for these terms.
     
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