Suppose we are given the map defined by . We would like to check if is linear. Take two vectors and and write

(1)

Next, write

(2)

Now observe that

(3)

Next we check the second condition and take a real number Write

(4)

Therefore, is a linear transformation.

Suppose is a linear transformation and we know

(5)

We would like to find using this information. Since is a linear transformation, we can write

(6)

This tells us that knowing where each elements of the basis is mapped by a linear transformation is enough to find the image of any other vector under

We are given the matrix

(7)

and would like to find the eigenvalues and associated eigenvectors of this matrix. We write

(8)

The eigenvalues of are and and are obtained by

(9)

Next, use the first eigenvalue to write and then reduce

(10)

Next,

(11)

implies . This is the eigenvector associated to the eigenvalue Next

(12)

We have

(13)

that is is the second eigenvector of and it’s associated to the eigenvalue