8.2. CONSTRUCTING THE MATRIX OF A LINEAR TRANSFORMATION
137
where ei is the vector which has zeros in every slot but the ith and a 1 in this slot. Then since T is linear, Tx
=
n X
xi T (ei )
i=1
= T ≡ A
| (e1 ) · · · | x1 .. .
x 1 | T (en ) ... | xn
xn and so you see that the matrix desired is obtained from letting the ith column equal T (ei ) . We state this as the following theorem. Theorem 8.2.1 Let T be a linear transformation from Fn to Fm . Then the matrix, A satisfying 8.1 is given by | | T (e1 ) · · · T (en ) | | where T ei is the ith column of A.
8.2.1
Rotations of R2
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. Example 8.2.2 Determine the matrix which represents the linear transformation defined by rotating every vector through an angle of θ. µ
¶ µ ¶ 1 0 and e2 ≡ . These identify the geometric vectors which point 0 1 along the positive x axis and positive y axis as shown. Let e1 ≡
e2 6
e1