I What is the definition of a matrix in function form?

[Nipon Waiyaworn]

My teacher told me to find the definition of matrix which is in function form, but haven't seen it.
The definition of matrix that I know is a rectangular arrangement of mn numbers, in m rows and n columns and enclosed within a bracket, but it is not right which my teacher wants.
I want to know what the definition of matrix which is in function form
help me please

 

[member 587159]

My teacher told me to find the definition of matrix which is in function form, but haven't seen it.
The definition of matrix that I know is a rectangular arrangement of mn numbers, in m rows and n columns and enclosed within a bracket, but it is not right which my teacher wants.
I want to know what the definition of matrix which is in function form
help me please

This question sounds stupid. You mustn't be asked to find a definition??

You can always try to google linear transformation and see if this helps you.

 

[Erland]

This question sounds stupid. You mustn't be asked to find a definition??

Why is it stupid?

 

[member 587159]

Because it is ambiguous.

 

[fresh_42]

My teacher told me to find the definition of matrix which is in function form, but haven't seen it.
The definition of matrix that I know is a rectangular arrangement of mn numbers, in m rows and n columns and enclosed within a bracket, but it is not right which my teacher wants.
I want to know what the definition of matrix which is in function form
help me please

You are both right. A matrix is a rectangular arrangement of numbers. At least normally, if we consider linear transformations and the matrix elements from an area like the real numbers. (In principle, one could arrange anything this way and call it a matrix, e.g. in image computations where they are pixels). So back to
$$
A = \begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{bmatrix}
$$
Now if we have an array $x = (\,x_1\; , \; x_2 \; , \; \ldots \; , \; x_n \,)$ we can define
$$
A \cdot x = ((Ax)_1, \ldots , (Ax)_m) = \begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{i1} & a_{i2} & \ldots & a_{in} \\ \vdots & \vdots && \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{bmatrix} \cdot \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_j \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} a_{11}x_1 + a_{12}x_2+ \ldots + a_{1n}x_n \\ a_{21}x_1+ a_{22}x_2+ \ldots + a_{2n}x_n \\ \quad \quad \quad \vdots \\ a_{i1}x_1 + a_{i2}x_2 + \ldots + a_{in}x_n \\ \quad \quad \quad \vdots \\ a_{m1}x_1+ a_{m2}x_2 + \ldots a_{mn}x_n \end{bmatrix}
$$
The arrays $x$ and $A\cdot x$ are called vectors and $A$ is a linear transformation. There is a bit more what should be said here about coordinates, dimensions and components, but basically this is it. You can look it up, e.g. on Wikipedia and the links there. But to get an impression what it's all about, choose $n=m=2$, some integers for $A$, and some examples for $x$. Then you can draw pairs of $x , Ax$ as arrows in a plane coordinate system with $x=x_1$ and $y=x_2$ axis where the arrows originate in the origin and end at the point $(x_1,x_2)$, resp. $((Ax)_1,(Ax)_2)$ and see what $A$ does to $x=(x_1,x_2)$.

 

[lavinia]

Another way to look at a matrix is that it is just a function on a doubly indexed set of points. The set of points can be abstract. It need not be a set of numbers or anything else. A function on this set just assigns a number to each of its elements.

This idea is the same as the idea of a function on any set for instance a function on the real line. The function $f(x) = x^2$ assigns a number to each point on the line. The matrix $\begin{bmatrix} 1& 5\\-7 & 0 \end{bmatrix}$ is the function $f(a_{11}) = 1$ $f(a_{12}) = 5$ $f(a_{21}) = -7$ $f(a_{22}) = 0$

 

[WWGD]

A "matrix function" can represent either a linear map or a bilinear map . For 3- 4- and higher linear maps you usually use tensors.

 

[I like Serena]

A matrix A is associated with the function given by $\mathbf x\mapsto A\cdot \mathbf x$.

 

[Hawkeye18]

I think your teacher meant the definition of a matrix as a function with domain being the set of pairs of integers $(j,k): 1\le j \le m, 1\le k\le n$. In this definition the value of this function at a point $(j,k)$ is $a_{j,k}$.

That is a general point of view in abstract mathematics, that everything (or almost everything) is a function. For example, a sequence is a function with domain being the set of natural numbers, and one can write $a(n)$ instead of $a_n$.

 

[WWGD]

Just to generalize a bit from what Hawkeye said, a matrix may be seen as a function in other ways: as an incidence matrix, it may be a binary-valued function ( i.e., function into $\{ 0,1\}$ with value 0 at $a_{i,j}$ if there is no edge between vertices $i,j$, similarly with Stochastic matrices, assigning to a pair of states $i,j$ the transition probability between $i,j$. etc.