Identity Maps: Decoding the Basics and Beyond

Twins are always a source of joy for people, especially when they are identical. Many couples are thrilled to discover they are having twins because they can dress them alike. However, despite their similar appearance, twins often have distinct personalities. Identity maps, on the other hand, are like twins that are identical both on the outside and inside, with no differences in personalities.

Understanding Identity Maps

An identity map is a concept in Linear Algebra, also known as an identity function, relation, operator, or transformation. These terms can be used interchangeably as we delve deeper into the topic.

In mathematics, a map illustrates the relationship between elements of two sets. Therefore, an identity map showcases the connection between elements from different sets.

Essentially, an identity map is a function that outputs the same value as its input.

For instance, the function

$$\begin{gathered} f\left(2\right)=2 \\ f(-5)=-5 \\ f\left(a\right)=a \\ f\left(x\right)=x \end{gathered}$$

serves as an identity function.

The identity map is sometimes denoted as Id(x) = x.

Properties of Identity Maps

Identity maps have a couple of key properties:

1. The elements in the domain and co-domain of the map are the same (it returns the value of its input).
2. The graph of an identity function is a straight line with a slope of 1.

Understanding Identity Maps in Linear Algebra

An identity map in linear algebra is represented by an identity matrix. This matrix is a square matrix where the diagonal elements are 1, and all other elements are 0.

Here are examples of a 2 x 2 and a 3 x 3 identity matrix:

A 2 x 2 identity matrix - $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

A 3 x 3 identity matrix - $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

The unique property of identity matrices is that when multiplied by themselves, they result in the same matrix. This holds true regardless of the matrix's dimensions.

Let's explore some examples to better understand this concept.

What happens when you square a $2\times 2$ identity matrix? And what about a $4\times 4$ identity matrix?

Answer:

A $2\times 2$ identity matrix is:

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Squaring the matrix above results in

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

A $4\times 4$ identity matrix is

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

Squaring the matrix above results in

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\times \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

When an identity matrix is squared, the result is always the identity matrix. This is why it is called an identity matrix.

You can find details on matrix multiplication in our article Operations with matrices

Understanding Identity Maps, Functions, and Transformations

As mentioned, the term "identity maps" is often used interchangeably with "identity functions" and "identity transformations" in the field of Mathematics.

Key Takeaways on Identity Maps

• The term "identity map" is synonymous with "identity function", "identity relation", "identity operator", and "identity transformation".
• The domain and co-domain of the map contain the same elements.
• The graph of an identity function forms a straight line.
• The identity map is associated with a matrix known as the identity matrix.
• The identity matrix is characterized by ones along the diagonal and zeros elsewhere.