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Thread: matrix

  1. #1
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    matrix

    How matrix can represent linear transformation?
    I need to take linear transofmration by matrix, How I do it?
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  2. #2
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    Re: matrix

    I don't understand your question. A matrix frequently represents a linear transformation. For example, the transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$ given by $T(x,y) = (2x+3y,-x+y)$ is represented by the matrix $\displaystyle \begin{bmatrix}2&3 \\ -1 & 1\end{bmatrix}$.
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    Re: matrix

    Thanks, You answer me as I meant.
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    Re: matrix

    What is the definition of T in the notion of you?
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  5. #5
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    Re: matrix

    Quote Originally Posted by policer View Post
    What is the definition of T in the notion of you?
    https://mathbitsnotebook.com/Algebra...valuation.html
    Check out this link on Function Notation. It should help you.
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  6. #6
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    Re: matrix

    I know f(x) notation.
    You renew me a new notation.
    Are There more notations to T?
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  7. #7
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    Re: matrix

    Quote Originally Posted by policer View Post
    I know f(x) notation.
    You renew me a new notation.
    Are There more notations to T?
    A function can be any letter. $f$, $T$, or any other letter you want to use. Note in the link I provided, they give several examples of this. A linear transformation is a particular type of function. I assumed you knew the definition of a linear transformation since you started this post asking about it, but here is that definition, as well (it is also called a linear map):
    https://en.wikipedia.org/wiki/Linear_map
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  8. #8
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    Re: matrix

    A little more questions:
    1.Why transformation is a function? Is there a proof to it?
    2. Are there transformations which are not functions?
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  9. #9
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    Re: matrix

    Quote Originally Posted by policer View Post
    A little more questions:
    1.Why transformation is a function? Is there a proof to it?
    2. Are there transformations which are not functions?
    1. Because a function is a general term and a transformation is a specific term. It is a specific type of function. A function takes an input and gives one output. A transformation takes an input and gives one output. Therefore, all transformations are functions, but not all functions are transformations.
    2. No.
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  10. #10
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    Re: matrix

    Quote Originally Posted by policer View Post
    A little more questions:
    1.Why transformation is a function? Is there a proof to it?
    2. Are there transformations which are not functions?
    Do you know what those words you are using mean? Do you know the definitions0 of "function" and "transformation"?

    A "transformation", from set A to set B, is any way of associating a member of B with a member of A. A more technical definition is that a transformation is a any subset of A x B, the set of all ordered pairs where the first member is from set A and the second is from set B.

    (Did you have "linear transformation" in mind?)

    A "function", from A to B, is a transformation such that to a given member of A there is no more that one member of B associated with it. A "function" is a subset of A x B such that for any x in A, there is no more than one pair having x as the first member.

    The formula $\displaystyle x^2+ y^2= 2$ can be interpreted as the transformation containing all pairs, (x, y), that satisfy that formula. Both (1, 1) and (1, -1) are in that set. Since there are two pairs containing the same first member, that is N0T a function.

    The formula $\displaystyle y= x^2$ can can be interpreted as the function containing all pair $\displaystyle (x, y)= (x, x^2$. For any x, $\displaystyle x^2$ is a single number, there is only one pair containing any specific x as first member.

    The set of all functions is a subset of the set of all transformations. Any function is a transformation but there exist transformations that are not functions.

    (If you mean "linear transformation", perhaps what SlipEternal has I mind, it is the other way around. A linear transformation is a function of the very simple form y= ax for some constant a. Every linear transformation is a function but not every function is a linear transformation.)
    Last edited by HallsofIvy; Apr 14th 2018 at 10:28 AM.
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  11. #11
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    Re: matrix

    1.Can you give a function that is not transformation except:
    f(x) = 2, for example. (I thnk - If I wrong, I will be happy to fix it).
    2. Can you give an example to transfomration that isn't a function?
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  12. #12
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    Re: matrix

    Quote Originally Posted by policer View Post
    1.Can you give a function that is not transformation except:
    f(x) = 2, for example. (I thnk - If I wrong, I will be happy to fix it).
    2. Can you give an example to transfomration that isn't a function?
    I do not recognize the definition that HallsofIvy was using. Here is the definition of transformation that I was taught:

    Transformation has a special meaning in math. How to reflect, translate, rotate in math...

    1. A function that is not a transformation: The Conway Base 13 Function is a function that is not a transformation. This is just one of my favorite functions. There are much easier functions that are not transformations (by the definition given in that article). For example: $f(x,y) = (x^2y,xy^2)$
    2. As far as I know, transformations are subsets of functions.
    Last edited by SlipEternal; Apr 21st 2018 at 08:19 AM.
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