A First Course in Linear Algebra, 2017a

5.3. Properties of Linear Transformations 279
Definition 5.16: Composition of Linear Transformations
Let T : R k ↦→ R n and S : R n ↦→ R m be linear transformations. Then the composite of S and T is
S ◦ T : R k ↦→ R m
The action of S ◦ T is given by
(S ◦ T)(⃗x)=S(T (⃗x)) for all⃗x ∈ R k
Notice that the resulting vector will be in R m . Be careful to observe the order of transformations. We
write S ◦ T but apply the transformation T first, followed by S.
Theorem 5.17: Composition of Transformations
Let T : R k ↦→ R n and S : R n ↦→ R m be linear transformations such that T is induced by the matrix
A and S is induced by the matrix B. ThenS ◦ T is a linear transformation which is induced by the
matrix BA.
Consider the following example.
Example 5.18: Composition of Transformations
Let T be a linear transformation induced by the matrix
[ ] 1 2
A =
2 0
and S a linear transformation induced by the matrix
[ ]
2 3
B =
0 1
[ 1
Find the matrix of the composite transformation S ◦ T. Then, find (S ◦ T)(⃗x) for ⃗x =
4
]
.
Solution. By Theorem 5.17, the matrix of S ◦ T is given by BA.
[ ][ ] [ 2 3 1 2 8 4
BA =
=
0 1 2 0 2 0
]
To find (S ◦ T)(⃗x), multiply⃗x by BA as follows
[ ][ 8 4 1
2 0 4
] [ 24
=
2
]
To check, first determine T (⃗x): [ 1 2
2 0
][ 1
4
] [ ] 9
=
2