Sometimes you need to find a matrix which represents a given linear transformation which is described in geometrical terms. The idea is to produce a matrix which you can multiply a vector by to get the same thing as some geometrical description. A good example of this is the problem of rotation of vectors discussed above. Consider the problem of rotating through an angle of θ.
Example 5.3.1 Determine the matrix which represents the linear transformation defined by rotating every vector through an angle of θ.
Let e_{1} ≡
From the above, you only need to find Te_{1} and Te_{2}, the first being the first column of the desired matrix, A and the second being the second column. From the definition of the cos,sin the coordinates of T(e_{1}) are as shown in the picture. The coordinates of T(e_{2}) also follow from simple trigonometry. Thus

Therefore, from Theorem 5.1.2,

For those who prefer a more algebraic approach, the definition of

Example 5.3.2 Find the matrix of the linear transformation which is obtained by first rotating all vectors through an angle of ϕ and then through an angle θ. Thus you want the linear transformation which rotates all angles through an angle of θ + ϕ.
Let T_{θ+ϕ} denote the linear transformation which rotates every vector through an angle of θ + ϕ. Then to get T_{θ+ϕ}, you could first do T_{ϕ} and then do T_{θ} where T_{ϕ} is the linear transformation which rotates through an angle of ϕ and T_{θ} is the linear transformation which rotates through an angle of θ. Denoting the corresponding matrices by A_{θ+ϕ}, A_{ϕ}, and A_{θ}, you must have for every x

Consequently, you must have
You do not have to stop with two dimensions. You can consider rotations and other geometric concepts in any number of dimensions. This is one of the major advantages of linear algebra. You can break down a difficult geometrical procedure into small steps, each corresponding to multiplication by an appropriate matrix. Then by multiplying the matrices, you can obtain a single matrix which can give you numerical information on the results of applying the given sequence of simple procedures. That which you could never visualize can still be understood to the extent of finding exact numerical answers. Another example follows.
Example 5.3.3 Find the matrix of the linear transformation which is obtained by first rotating all vectors through an angle of π∕6 and then reflecting through the x axis.
As shown in Example 5.3.2, the matrix of the transformation which involves rotating through an angle of π∕6 is

The matrix for the transformation which reflects all vectors through the x axis is

Therefore, the matrix of the linear transformation which first rotates through π∕6 and then reflects through the x axis is
