How matrix can represent linear transformation?
I need to take linear transofmration by matrix, How I do it?
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 $\displaystyle \begin{bmatrix}2&3 \\ -1 & 1\end{bmatrix}$.
https://mathbitsnotebook.com/Algebra...valuation.html
Check out this link on Function Notation. It should help you.
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
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.
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.)
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.