A First Course in Linear Algebra, 2017a

5.5. One to One and Onto Transformations 287
Exercise 5.4.15 Find the matrix of the linear transformation which rotates every vector in R 3 counter
clockwise about the z axis when viewed from the positive z axis through an angle of 30 ◦ and then reflects
through the xy plane.
[ ] a
Exercise 5.4.16 Let ⃗u = be a unit vector in R
b
2 . Find the matrix which reflects all vectors across
this vector, as shown in the following picture.
⃗u
[ ] a
Hint: Notice that =
b
axis. Finally rotate through θ.
[ cosθ
sinθ
5.5 One to One and Onto Transformations
]
for some θ. First rotate through −θ. Next reflect through the x
Outcomes
A. Determine if a linear transformation is onto or one to one.
Let T : R n ↦→ R m be a linear transformation. We define the range or image of T as the set of vectors
of R m which are of the form T (⃗x) (equivalently, A⃗x)forsome⃗x ∈ R n . It is common to write T R n , T (R n ),
or Im(T ) to denote these vectors.
Lemma 5.28: Range of a Matrix Transformation
Let A be an m × n matrix where A 1 ,···,A n denote the columns of A. Then, for a vector⃗x =
in R n ,
A⃗x =
n
∑ x k A k
k=1
Therefore, A(R n ) is the collection of all linear combinations of these products.
⎡
⎢
⎣
⎤
x 1
.. ⎥
. ⎦
x n
Proof. This follows from the definition of matrix multiplication.
♠
This section is devoted to studying two important characterizations of linear transformations, called
one to one and onto. We define them now.