Can someone explain to me what a matrix is in simple words?

[Solid Snake]

Ok so officially a matrix is a rectangular array of numbers, symbols, etc arranged in rows and columns that is treated in certain prescribed ways.

But that doesn't help me understand a darn thing. From what I understand, a matrix is a math tool that can help you solve linear systems, represent tables, and represent vectors (that's another whole confusion in it of itself for me).

Like when I'm working with real life, where do I see something and say, "oh, that's a matrix!"? I can go outside and see real life events and say "oh, I can represent that as a function!". How does one go about doing that with matrices?

I personally wish someone would explain to me what a matrix is in simple layman's terms.

 

[paisiello2]

A matrix simply is formed from a series (or system) of equations. This is where it can be used directly to solve linear systems.

A real life application would be if you wanted to model mathematically a real world situation such as the structure of an aircraft. You would make some approximations in your model to come with your system of equations and then solve this system to come up with the approximate behavior of the aircraft.

 

[Solid Snake]

A matrix simply is formed from a series (or system) of equations. This is where it can be used directly to solve linear systems.

A real life application would be if you wanted to model mathematically a real world situation such as the structure of an aircraft. You would make some approximations in your model to come with your system of equations and then solve this system to come up with the approximate behavior of the aircraft.

Thank you.

But what about vectors. How in god's good name do matrices get tied up with vectors and geometry?

I now understand that it is used to solve linear systems, but how is it used in vectors?

If someone could explain this and more in simple layman's terms, that would be great!

Thanks.

 

[scientific601]

I would say matrices are mathematical constructs. A way of organizing numbers so that subsequent computation can processed with well defined repetition sequences.

Whenever you want to work with a set of numbers... they can represent coefficients of equations, or just tables of employee ages, incomes, date of hiring, etc. The closest thing to a 'real life' matrix might be the sensor in a camera. A rectangularly organized set of components that convert light energy to electric signals. The abstraction of that, the mathematical equivalent would likely contain those values in voltage, or lumens.
An even more obvious example: a wall calendar. It's organized as a set of days, each row is a week.

As for vectors, geometry... again, a matrix would be used to hold values. X, Y, Z coordinates for example, then other matrices may be used to represent rotations in 3-D space.
A matrix is quite generic in that its contents can hold meaning of whatever you're trying to model.

 

[WWGD]

Ok so officially a matrix is a rectangular array of numbers, symbols, etc arranged in rows and columns that is treated in certain prescribed ways.

But that doesn't help me understand a darn thing. From what I understand, a matrix is a math tool that can help you solve linear systems, represent tables, and represent vectors (that's another whole confusion in it of itself for me).

Like when I'm working with real life, where do I see something and say, "oh, that's a matrix!"? I can go outside and see real life events and say "oh, I can represent that as a function!". How does one go about doing that with matrices?

I personally wish someone would explain to me what a matrix is in simple layman's terms.

You are correct, it is an array of numbers, but its meaning/properties depend on the context. A matrix can serve to describe many different things: systems of linear equations, adjacency graphs , Markov chains, etc.

 

[disregardthat]

You may think of an n x m (n rows, m columns) matrix A as a linear function $A : \mathbb{R}^m \to \mathbb{R}^n$, $A(x) = Ax$ is just multiplication by the matrix with a vector (a special form of matrix multiplication since vectors in $\mathbb{R}^m$ are m x 1 matrices). Just like linear functions $f : \mathbb{R} \to \mathbb{R}$ such that $f(x) = ax$, the matrix (or linear function) A has a constant derivative. In this sense it is a generalization of ordinary linear functions. The derivative will be another matrix, called the jacobi-matrix (in fact, the same matrix). While general differentiable functions $f : \mathbb{R}^m \to \mathbb{R}^n$ have non-constant jacobi-matrices (derivatives), linear functions are exactly those with constant jacobi-matrices.

 

[geoffrey159]

Matrices are representations of linear mappings
Any linear mapping $f$ from $E\rightarrow F$, where $E$ and $F$ are finite dimensional vector spaces, can be represented by a matrix.
If $(e_1,...,e_n)$ is a basis of $E$, $f$ is entirely determined by $f(e_1),...,f(e_n)$.
Each column of a matrix are the coordinates of $f(e1),...,f(e_n)$ expressed in a basis $(f_1,...,f_p)$ of $F$
The matrix product represents the the composition of two linear mappings ( $f\circ g$, with $f,g$ linear mappings).

 

[WWGD]

But a matrix can represent things/situations other than linear maps between f.d spaces ( I think modules are the greatest level of generality). Matrices can also describe systems of linear equations, adjacency relations in a graph, or a Markov process, etc. The meaning of a matrix, what it represents, depends on the context in which they appear.

 

[geoffrey159]

You must be right, and I don't have the knowledge to elaborate about what you say. This is how I have been taught about matrices, and I thought it could be a starting point for the OP.

 

[geoffrey159]

Matrices are representations of linear mappings
Any linear mapping $f$ from $E\rightarrow F$, where $E$ and $F$ are finite dimensional vector spaces, can be represented by a matrix.
If $(e_1,...,e_n)$ is a basis of $E$, $f$ is entirely determined by $f(e_1),...,f(e_n)$.
Each column of a matrix are the coordinates of $f(e1),...,f(e_n)$ expressed in a basis $(f_1,...,f_p)$ of $F$
The matrix product represents the the composition of two linear mappings ( $f\circ g$, with $f,g$ linear mappings).

As an example, take $E$ as the vector space of polynomials of degree less or equal to $n$.
And take $f : E \rightarrow E$ as the linear mapping defined by $f : P(X) \rightarrow P(X+1)$. What is the matrix of $f$ relative to the basis $(1,X,X^2,..., X^n)$ of $E$ ?
You know that each column are the coordinates of $f(X^k)_{k=0...n}$ expressed in the basis $(1,X,X^2,..., X^n)$.
You've got that $f(X^k) = (X+1)^k = \sum_{i = 0}^k C_k^i X^i$.

So the matrix of $f$ is something like $A = \begin{pmatrix} 1 & 1 & \cdots & 1\\ 0 & C_1^1 & \cdots & C_n^1 \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & C_n^n \end{pmatrix}$

If you have got it, then what is $A^{-1}$ ?