(c)2015 UM Math Dept
licensed under a Creative Commons
By-NC-SA 4.0 International License.
Math 217: §2.2 Linear Transformations and Geometry
Professor Karen Smith
Key Definition: A linear transformation T : Rn → Rm is a map (i.e., a function) from Rn to
Rm satisfying the following:
• T (~x + ~y ) = T (~x) + T (~y ) for all ~x, ~y ∈ Rn (that is, “T respects addition”).
• T (a~x) = aT (~x) for all a ∈ R and ~x ∈ Rn (that is, “T respects scalar mutliplication”).
C. Let S : R2 → R2 be dilation by a factor of three.
1. Give a geometric reason that S is a linear transformation using the definition.
2. What is the associated matrix A so that S(~v ) = A~v ?
3. What about dilation (or contration) by an arbitrary factor?
Solution note: It is easy to check that scaling preserves addition and scalar
multipli
λ 0
cation of vectors (thinking of them as directed arrows). The matrix is
, where
0 λ
λ = 3 in (1) or any scale factor.
D. Let L : R2 → R2 be rotation in the counter-clockwise direction by 90◦ (fixing the origin).
1. Give a geometric explanation why L is a linear transformation using the definition.
2. What is the associated matrix A so that L(~v ) = A~v ?
3. What about rotation through an arbitrary angle θ? To write the matrix, you need to remember your high school trig.
Solution note: Again, adding ”arrows” head to tail as in Math 215 is the same before
or
of any angle. Same for scaling. So this is linear. The matrix is
after a rotation
cosθ −sinθ
.
sinθ cosθ
E. Let M : R2 → R2 be reflection over the x-axis.
1. Show that M is linear by writing down a formula for it explicitly.
2. What about reflection over the line y = x? Is this a linear tranformation? If so, find its
matrix.
x
x
1 0
7→
. The matrix is
.
y
−y
0 −1
Reflection over the line y = x is also linear, as it just swaps the x and y coordinates,
0 1
and you can check that this respects addition and scalar mult. The matrix is
.
1 0
Solution note: Yup, linear. A formula is
F. Let Q : R2 → R2 be the transformation that stretches vertically by a factor of two and contracts
horizontally by a factor of 3.
1. Show that Q is linear by writing down a formula for it explicitly.
2. What about arbitrary (but different) scale factors vertically and horizontally? What happens
if they are negative?
x
1/3x
Solution note: The formula is
7→
. You can verify using this formula that
y
2y
0
0
1/3 0
x
x
x
x
. Any scale
) + Q( 0 ). The matrix is
+ 0 ) = Q(
Q is linear: Q(
0 2
y
y
y
y
factors are fine, including negative, which geometrically involves a reflection.
H. Bonus: Think geometrically: Do you think that reflection over an arbitrary line through the
origin is a linear transformation? Can you write down its matrix?
Solution note: The answer to this is YES. You’ll get a worksheet on this soon.