A First Course in Linear Algebra, 2017a

5.2. The Matrix of a Linear Transformation I 273
2. The columns of the matrix for T are defined above as T (⃗e i ). It follows that T (⃗e i )=proj ⃗u (⃗e i ) gives
the i th column of the desired matrix. Therefore, we need to find
( ) ⃗ei •⃗u
proj ⃗u (⃗e i )= ⃗u
⃗u •⃗u
For the given vector ⃗u , this implies the columns of the desired matrix are
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1
1
⎣ 2 ⎦, 2 1
⎣ 2 ⎦, 3 1
⎣ 2 ⎦
14 14 14
3 3 3
which you can verify. Hence the matrix of T is
⎡
1 2 3
1
⎣ 2 4 6
14
3 6 9
⎤
⎦
♠
Exercises
Exercise 5.2.1 Consider the following functions which map R n to R n .
(a) T multiplies the j th component of⃗x by a nonzero number b.
(b) T replaces the i th component of⃗x with b times the j th component added to the i th component.
(c) T switches the i th and j th components.
Show these functions are linear transformations and describe their matrices A such that T (⃗x)=A⃗x.
Exercise 5.2.2 You are given a linear transformation T : R n → R m and you know that
T (A i )=B i
where [ ] −1
A 1 ··· A n exists. Show that the matrix of T is of the form
[ ][ ] −1
B1 ··· B n A1 ··· A n
Exercise 5.2.3 Suppose T is a linear transformation such that
⎡
T ⎣
⎤
1
2 ⎦ =
⎡ ⎤
5
⎣ 1 ⎦
−6 3