Identity Map

Plot the graph for the following identity function.

Answer:

Plotting the graph gives:

From the graph, you can see that we have a straight line. We take the input as x and the output as y, forming the line. That is, (1, 1), (2, 2), (3, 3), and (4, 4).

Use the table below to plot a graph of the function f(x) and determine if the function is an identity function.

Which of the following image does NOT represent an identity map?

Answer:

This can be a bit tricky, so you have to look closely. If you observe image A, you will see that a maps to a, b maps to b, c maps to c, and d maps to d. The output is an exact image of the input, meaning it is an identity map.

If you observe the second image, a maps to c, b maps to d, c maps to b, and d maps to a. This means that it is not an identity map because the elements do not map to themselves.

From the third image, it's apparent that all elements map to themselves. So, it's an identity map.

So, the answer to the question is B because the elements do not map to themselves.

Prove that $f\left(4x\right)=4x$ is an identity function and draw the identity map.

Answer:

For the function to be identical, the input and output must be identical. So, what we will do here is to plug in different values for x and see if the input and the output will be the same.

If x = 1, $f(4\times 1)=4\times 1=4$

If x = 2, $f(4\times 2)=4\times 2=8$

If x = 4, $f(4\times 4)=4\times 4=16$

If x = 5, $f(4\times 5)=4\times 5=20$

We can see that no matter the value of x, the output and the input will still be equal. This means that the function f is an identical map. The figure below shows the identity map.

What is the result when you square a $2\times 2$ identity matrix? What about if you square 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 yields

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

A $4\times 4$ identity matrix is

Squaring the matrix above yields

As you can see, when an identity matrix is multiplied by itself, the result is the identity matrix. This is why it is related to an identity map.