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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”).
1
4
0
−2
A. Suppose that
−→
is a linear transformation. Suppose T (
)=
and T (
)=
.
0
2
1
−2
1
? Find it, using only the definition of linear transformation
1. Do we also know the value of T
1
2
2
?
? T
given above. What about T
1
0
R2
T
R2
2. Do we know the value of T on any linear combination a~e1 + b~e2 where ei are the standard
unit column vectors in R2 ? Find it, using only the definition of linear transformation given
above.
a
)? Prove it.
3. What is T (
b
4. Find the matrix A such that T ~x = A~x.
5. What does your matrix have to do with T (~e1 ) and T (~e2 )?
6. Can you state a general conjecture? Can you prove your conjecture?
B. Let ~e1 , . . . , ~en be the standard unit vectors for Rn .
1. If we know the values of a linear transformation T : Rn → Rd on each ~ei , do we know the
value for any ~x ∈ Rn ? Why? Discuss with your tablemates.
2. Prove that T (~x) = A~x where A is the d × n matrix formed by the vectors T (~e1 ), . . . T (~en ).
This is a crucial idea. Be sure you understand exactly how a linear transformation
can be described using matrix multiplication, and how to get the matrix.
Linear transformations in geometry
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?
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.
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.
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?
G. Find the matrix for a vertical shear by 2. What about a horizontal shear by λ? What happens
if λ is negative? Use the “standard L” as in the book to analyze this linear transformation.
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?