Study Guide: Linear Transformations
1. Functions
In advanced mathematics a function may take inputs from any set X, and its outputs may
be elements of any set Y . If f is a function with inputs from X and outputs from Y , we write
it f : X → Y .
A transformation of Rn is any function T : Rn → Rn . That is, a transformation of Rn is a
function whose inputs and outputs are points in Rn .
2. Linear Transformations
A linear transformation of Rn is a transformation T : Rn → Rn of the form
T (x) = Ax
where A is an n × n matrix whose entries are constants.
For example, a linear transformation of R2 is a function of the form
a b x
T (x, y) =
or equivalently
T (x, y) = (ax + by, cx + dy).
c d y
Similarly, a linear

a

T (x, y) = d
g
transformation of R3 is a function of the form
 
b c
x


e f
y  or T (x, y) = (ax + by + cz, dx + ey + f z, gx + hy + iz).
h i
z
3. Finding the Matrix
Given a linear transformation T of R2 , you can often find the matrix for T using the following
procedure:
1. Compute T (1, 0). This is the first column of the matrix.
2. Compute T (0, 1). This is the second column of the matrix.
A similar procedure works for linear transformations of R3 .
In some cases it’s not obvious what T (1, 0) or T (0, 1) might be. If you are given (or can figure
out) T (v) and T (w) for some other vectors v and w, you can solve for the entries of the matrix
by setting up a system of equations. For example, if T (1, 1) = (3, 7) and T (−1, 2) = (1, 4),
then we get
a b 1
3
a b −1
1
=
and
=
c d 1
7
c d
2
4
which gives a system of four equations involving a, b, c, and d.
4. Rotations
A counterclockwise rotation of R2 around the origin corresponds to the matrix
cos θ − sin θ
sin θ
cos θ
where θ is the angle of rotation. The eigenvalues of such a matrix are complex numbers:
λ = cos θ ± i sin θ.
In R3 , a rotation matrix has three eigenvalues:
λ = 1
and
λ = cos θ ± i sin θ
where θ is the angle of rotation. The eigenspace for λ = 1 is a line through the origin, and is
called the axis of rotation.
To find the angle for a rotation in R3 , you find the complex eigenvalues and then take the
inverse cosine of the real part. For example, if the complex eigenvalues are
√
1
3
λ =
±
i
2
2
then θ = cos−1 (1/2) = 60◦ .
5. Reflections
A reflection in R2 always has eigenvalues λ = 1 and λ = −1. The eigenspace for λ = 1 is the
line of reflection, and the eigenspace for λ = −1 is the perpendicular line.
A reflection in R3 always has eigenvalues λ = 1 (with multiplicity two) and λ = −1. The
eigenspace for λ = 1 is the plane of reflection, and the eigenspace for λ = −1 is the
perpendicular line.