1. ## matrix

How matrix can represent linear transformation?
I need to take linear transofmration by matrix, How I do it?

2. ## 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 $\displaystyle \begin{bmatrix}2&3 \\ -1 & 1\end{bmatrix}$.

3. ## Re: matrix

Thanks, You answer me as I meant.

4. ## Re: matrix

What is the definition of T in the notion of you?

5. ## Re: matrix

Originally Posted by policer
What is the definition of T in the notion of you?
https://mathbitsnotebook.com/Algebra...valuation.html

6. ## Re: matrix

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

7. ## Re: matrix

Originally Posted by policer
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

8. ## 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?

9. ## Re: matrix

Originally Posted by policer
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.

10. ## Re: matrix

Originally Posted by policer
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.)

11. ## 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?

12. ## Re: matrix

Originally Posted by policer
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.