A First Course in Linear Algebra, 2017a

268 Linear Transformations
Exercises
Exercise 5.1.1 Show the map T : R n ↦→ R m defined by T (⃗x)=A⃗x whereAisanm× n matrix and⃗x isan
m × 1 column vector is a linear transformation.
Exercise 5.1.2 Show that the function T ⃗u defined by T ⃗u (⃗v)=⃗v − proj ⃗u (⃗v) is also a linear transformation.
Exercise 5.1.3 Let ⃗u be a fixed vector. The function T ⃗u defined by T ⃗u ⃗v =⃗u +⃗v has the effect of translating
all vectors by adding ⃗u ≠⃗0. Show this is not a linear transformation. Explain why it is not possible to
represent T ⃗u in R 3 by multiplying by a 3 × 3 matrix.
5.2 The Matrix of a Linear Transformation I
Outcomes
A. Find the matrix of a linear transformation with respect to the standard basis.
B. Determine the action of a linear transformation on a vector in R n .
In the above examples, the action of the linear transformations was to multiply by a matrix. It turns
out that this is always the case for linear transformations. If T is any linear transformation which maps
R n to R m ,thereisalways an m × n matrix A with the property that
for all⃗x ∈ R n .
Theorem 5.5: Matrix of a Linear Transformation
T (⃗x)=A⃗x (5.1)
Let T : R n ↦→ R m be a linear transformation. Then we can find a matrix A such that T (⃗x)=A⃗x. In
this case, we say that T is determined or induced by the matrix A.
Here is why. Suppose T : R n ↦→ R m is a linear transformation and you want to find the matrix defined
by this linear transformation as described in 5.1. Note that
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x 1 1 0
0
x 2
n ⃗x = ⎢ ⎥
⎣ . ⎦ = x 1 ⎢ ⎥
⎣
0. ⎦ + x 2 ⎢ ⎥
⎣
1. ⎦ + ···+ x n ⎢ ⎥
⎣
0. ⎦ = ∑ x i ⃗e i
i=1
x n 0 0
1
where ⃗e i is the i th column of I n , that is the n × 1 vector which has zeros in every slot but the i th and a 1 in
this slot.
Then since T is linear,
T (⃗x) =
n
∑
i=1
x i T (⃗e i )