8.5. SOME EXAMPLES OF LINEAR FUNCTIONS ON Rn 139

Example 8.4.8 Let

A =

 2 1 31 5 −33 −3 7

 .

Then A is symmetric.

Example 8.4.9 Let

A =

 0 1 3−1 0 2−3 −2 0

Then A is skew symmetric.

8.5 Some Examples of Linear Functions on Rn

There are many examples of linear functions and we give a couple next.

8.5.1 Rotations in R2

Sometimes you need to find a matrix which represents a given linear transformation whichis described in geometrical terms. The idea is to produce a matrix which you can multiplya vector by to get the same thing as some geometrical description. A good example of thisis the problem of rotation of vectors discussed above. Consider the problem of rotatingthrough an angle of θ .

Example 8.5.1 Determine the matrix which represents the linear transformation definedby rotating every vector through an angle of θ .

Let e1 ≡

(10

)and e2 ≡

(01

). These identify the geometric vectors which point

along the positive x axis and positive y axis as shown.

e1

e2

θ

θ

(cos(θ),sin(θ))(−sin(θ),cos(θ))T (e1)

T (e2)

From the above, you only need to find Te1 and Te2, the first being the first column ofthe desired matrix, A and the second being the second column. From the definition of the