Download Introduction to Linear Transformation

Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Introduction to Linear Transformation
Math 4A – Xianzhe Dai
UCSB
April 14 2014
Based on the 2013 Millett and Scharlemann Lectures
1/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
System of Linear Equations:
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
..
.
am1 x1 + am2 x2 + . . . + amn xn = bm
2/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
System of Linear Equations:
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
..
.
am1 x1 + am2 x2 + . . . + amn xn = bm
can be written as matrix equation A~x = ~b.
2/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
System of Linear Equations:
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
..
.
am1 x1 + am2 x2 + . . . + amn xn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a
“function/map/transformation”, T (~x ) = A~x . T maps/transforms
a vector ~x to another vector A~x .
2/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
System of Linear Equations:
a11 x1 + a12 x2 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn = b2
..
.
am1 x1 + am2 x2 + . . . + amn xn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a
“function/map/transformation”, T (~x ) = A~x . T maps/transforms
a vector ~x to another vector A~x .
Two very nice properties it enjoys are
T (~u + ~v ) = A(~u + ~v ) = A~u + A~v
T (c~u )
=
A(c~u )
=
cA~u
= T (~u ) + T (~v )
=
cT (~u )
2/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
3/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
Example: Let T : R1 → R1 be defined by T (x) = 5x.
3/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
Example: Let T : R1 → R1 be defined by T (x) = 5x.
For any u, v ∈ R 1 ,
T (u + v ) = 5(u + v ) = 5u + 5v = T (u) + T (v ) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
3/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
Example: Let T : R1 → R1 be defined by T (x) = 5x.
For any u, v ∈ R 1 ,
T (u + v ) = 5(u + v ) = 5u + 5v = T (u) + T (v ) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
So T is a linear transformation.
3/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some notes:
Most functions are not linear transformations.
4/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some notes:
Most functions are not linear transformations. For example:
cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ).
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Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some notes:
Most functions are not linear transformations. For example:
cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ).
For any linear transformation T (~0) = ~0 (this rules out
function f (x) = x + 5):
4/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some notes:
Most functions are not linear transformations. For example:
cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ).
For any linear transformation T (~0) = ~0 (this rules out
function f (x) = x + 5): Take c = 0, then
T (~0) = T (0 · ~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors
~u , ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v ) = aT (~u ) + bT (~v )
.
4/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some notes:
Most functions are not linear transformations. For example:
cos(x + y ) 6= cos(x) + cos(y ). Or (2x)2 6= 2(x 2 ).
For any linear transformation T (~0) = ~0 (this rules out
function f (x) = x + 5): Take c = 0, then
T (~0) = T (0 · ~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors
~u , ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v ) = aT (~u ) + bT (~v )
.
Example
4/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x .
We have already seen that T has what it takes:
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x .
We have already seen that T has what it takes:
For any vectors ~u , ~v ∈ Rn ,
T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v )
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x .
We have already seen that T has what it takes:
For any vectors ~u , ~v ∈ Rn ,
T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R,
T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ).
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x .
We have already seen that T has what it takes:
For any vectors ~u , ~v ∈ Rn ,
T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R,
T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ).
A linear transformation defined by a matrix is called a matrix
transformation.
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x ) = A~x .
We have already seen that T has what it takes:
For any vectors ~u , ~v ∈ Rn ,
T (~u + ~v ) = A(~u + ~v ) = A~u + A~v = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R,
T (c~u ) = A(c~u ) = c(A~u ) = cT (~u ).
A linear transformation defined by a matrix is called a matrix
transformation.
Important Fact Conversely any linear transformation is associated
to a matrix transformation (by using bases).
5/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Mona Lisa transformed
6/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix transformations are important and are also cool!
7/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix transformations are important and are also cool!
Example 1, a shear: Consider the matrix transformation
T : R2 → R2 given by the 2 × 2 matrix
1 32
A=
0 1
7/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix transformations are important and are also cool!
Example 1, a shear: Consider the matrix transformation
T : R2 → R2 given by the 2 × 2 matrix
1 32
A=
0 1
x1
0
x1
0
For any horizontal vector ~x =
T (~x ) = A~x =
1 23
0 1
=
x1 + 32 · 0
0 · x1 + 1 · 0
=
x1
0
7/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix transformations are important and are also cool!
Example 1, a shear: Consider the matrix transformation
T : R2 → R2 given by the 2 × 2 matrix
1 32
A=
0 1
x1
0
x1
0
For any horizontal vector ~x =
T (~x ) = A~x =
1 23
0 1
=
x1 + 32 · 0
0 · x1 + 1 · 0
=
x1
0
So T is the identity on horizontal vectors.
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Linear Transformations
Pictorial examples
For any vertical vector ~x =
0
x2
Matrix Is Everywhere
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Linear Transformations
Pictorial examples
0
x2
0
x2
For any vertical vector ~x =
T (~x ) = A~x =
1 32
0 1
Matrix Is Everywhere
=
1 · 0 + 23 · x2
0 · 0 + 1 · x2
=
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Linear Transformations
Pictorial examples
0
x2
0
x2
For any vertical vector ~x =
T (~x ) = A~x =
1 32
0 1
Matrix Is Everywhere
=
1 · 0 + 23 · x2
0 · 0 + 1 · x2
=
0
x2
+
3
2 x2
0
8/24
Prelude
Linear Transformations
Pictorial examples
0
x2
0
x2
For any vertical vector ~x =
T (~x ) = A~x =
1 32
0 1
Matrix Is Everywhere
=
1 · 0 + 23 · x2
0 · 0 + 1 · x2
=
0
x2
+
So a vertical vector is pushed perfectly horizontally, a distance
times its length:
3
2 x2
0
3
2
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Prelude
Linear Transformations
Pictorial examples
0
x2
0
x2
For any vertical vector ~x =
T (~x ) = A~x =
1 32
0 1
Matrix Is Everywhere
=
1 · 0 + 23 · x2
0 · 0 + 1 · x2
=
0
x2
+
So a vertical vector is pushed perfectly horizontally, a distance
times its length:
x2
(0, 1)
(3/2, 1) (2, 1)
3
2 x2
0
3
2
(2+3/2, 1)
x1
(2, 0)
8/24
Prelude
Linear Transformations
Pictorial examples
Example 2, scaling:
Use
A=
2 0
0 12
Matrix Is Everywhere
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Prelude
Linear Transformations
Pictorial examples
Example 2, scaling:
Use
For any vector ~x =
T (~x ) = A~x =
x1
x2
2 0
0 12
A=
x1
x2
2 0
0 12
=
Matrix Is Everywhere
2x1 + 0 · x2
0 · x1 + 21 · x2
=
2x1
x2
2
9/24
Prelude
Linear Transformations
Pictorial examples
Example 2, scaling:
Use
For any vector ~x =
T (~x ) = A~x =
x1
x2
2 0
0 12
A=
x1
x2
2 0
0 12
=
Matrix Is Everywhere
2x1 + 0 · x2
0 · x1 + 21 · x2
=
2x1
x2
2
So T stretches horizontally and contracts vertically:
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Prelude
Linear Transformations
Pictorial examples
Example 2, scaling:
Use
For any vector ~x =
T (~x ) = A~x =
x1
x2
2 0
0 12
A=
x1
x2
2 0
0 12
=
Matrix Is Everywhere
2x1 + 0 · x2
0 · x1 + 21 · x2
=
2x1
x2
2
So T stretches horizontally and contracts vertically:
x2
(0, 1)
x1
(2, 0)
9/24
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Linear Transformations
Pictorial examples
Matrix Is Everywhere
Example 3, reflection through a line:
Use
0 1
A=
1 0
10/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Example 3, reflection through a line:
Use
0 1
A=
1 0
x1
x1
0 1
x1
0x1 + 1 · x2
x2
T(
)=A
=
=
=
x2
x2
1 0
x2
1 · x1 + 0 · x2
x1
10/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Example 3, reflection through a line:
Use
0 1
A=
1 0
x1
x1
0 1
x1
0x1 + 1 · x2
x2
T(
)=A
=
=
=
x2
x2
1 0
x2
1 · x1 + 0 · x2
x1
So T exchanges the two coordinates.
10/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Example 3, reflection through a line:
Use
0 1
A=
1 0
x1
x1
0 1
x1
0x1 + 1 · x2
x2
T(
)=A
=
=
=
x2
x2
1 0
x2
1 · x1 + 0 · x2
x1
So T exchanges the two coordinates. Looks like reflection through
the line x1 = x2 :
10/24
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Linear Transformations
Pictorial examples
Matrix Is Everywhere
Example 3, reflection through a line:
Use
0 1
A=
1 0
x1
x1
0 1
x1
0x1 + 1 · x2
x2
T(
)=A
=
=
=
x2
x2
1 0
x2
1 · x1 + 0 · x2
x1
So T exchanges the two coordinates. Looks like reflection through
the line x1 = x2 :
x2
(s, s)
reflect
tcelfer
x1
10/24
Prelude
Linear Transformations
Example 4, rotation:
Use
Pictorial examples
A=
cos θ − sin θ
sin θ cos θ
Matrix Is Everywhere
11/24
Prelude
Linear Transformations
Example 4, rotation:
Use
Pictorial examples
A=
cos θ − sin θ
sin θ cos θ
Matrix Is Everywhere
1
1
cos θ − sin θ
1
cos θ
T(
)=A
=
=
0
0
sin θ cos θ
0
sin θ
11/24
Prelude
Linear Transformations
Example 4, rotation:
Use
Pictorial examples
A=
cos θ − sin θ
sin θ cos θ
Matrix Is Everywhere
1
1
cos θ − sin θ
1
cos θ
T(
)=A
=
=
0
0
sin θ cos θ
0
sin θ
So horizontal unit vector is rotated c-clockwise an angle θ.
11/24
Prelude
Linear Transformations
Example 4, rotation:
Use
Pictorial examples
A=
cos θ − sin θ
sin θ cos θ
Matrix Is Everywhere
1
1
cos θ
T(
)=A
=
0
0
sin θ
So horizontal unit vector is rotated
− sin θ
1
cos θ
=
cos θ
0
sin θ
c-clockwise
an angle θ.
0
Similarly, for the vertical unit vector
,
1
11/24
Prelude
Linear Transformations
Pictorial examples
Example 4, rotation:
Use
A=
cos θ − sin θ
sin θ cos θ
Matrix Is Everywhere
1
1
cos θ
T(
)=A
=
0
0
sin θ
So horizontal unit vector is rotated
− sin θ
1
cos θ
=
cos θ
0
sin θ
c-clockwise
an angle θ.
0
Similarly, for the vertical unit vector
, so all of plane rotates:
1
R (v)
v
11/24
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Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
Answer: Just do matrix-vector multiplication A~u . The result is a
vector in Rm .
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
Answer: Just do matrix-vector multiplication A~u . The result is a
vector in Rm .
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~b ∈ Rm is given.
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
Answer: Just do matrix-vector multiplication A~u . The result is a
vector in Rm .
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~b ∈ Rm is given.
Find a vector ~u so that
T (~u ) = ~b.
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
Answer: Just do matrix-vector multiplication A~u . The result is a
vector in Rm .
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~b ∈ Rm is given.
Find a vector ~u so that
T (~u ) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u .
12/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u )?
Answer: Just do matrix-vector multiplication A~u . The result is a
vector in Rm .
Question: Suppose T : Rn → Rm is a matrix linear transformation.
Suppose A is the matrix of T and ~b ∈ Rm is given.
Find a vector ~u so that
T (~u ) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u .
Reminder: There may be
no solution or
exactly one solution or
a parameterized family of solutions
12/24
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Linear Transformations
Pictorial examples
Matrix Is Everywhere
iClicker question
Suppose T is a matrix linear transformation with matrix A below,
and we are seeking all vectors ~u so that T (~u ) = ~b.


 
1 2 3
6
~b = 7
A = 0 4 5
0 0 0
8
How many solutions are there?
A)
B)
C)
D)
Zero.
One
Infinity
Can’t tell
13/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
iClicker question
Suppose T is a matrix linear transformation with matrix A below,
and we are seeking all vectors ~u so that T (~u ) = ~b.


 
1 2 3
6
~b = 7
A = 0 4 5
0 0 0
8
How many solutions are there?
A)
B)
C)
D)
Zero.
One
Infinity
Can’t tell
 
6
What if ~b = 7?
0
13/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u ) = ~b.
14/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u ) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.
Could be no ~u , could be exactly one ~u , or could be a parametrized
family of such ~u ’s.
14/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u ) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.
Could be no ~u , could be exactly one ~u , or could be a parametrized
family of such ~u ’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merely
echelon form.
14/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u ) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.
Could be no ~u , could be exactly one ~u , or could be a parametrized
family of such ~u ’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merely
echelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ unique
solution.
There are free variables ⇐⇒ there is a parameterized family
of solutions whose dimension is the number of free variables.
14/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u ) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.
Could be no ~u , could be exactly one ~u , or could be a parametrized
family of such ~u ’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merely
echelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ unique
solution.
There are free variables ⇐⇒ there is a parameterized family
of solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy to
describe them.
14/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
15/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
For any linear transformation T (~0) = ~0
15/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
For any linear transformation T (~0) = ~0
T (a~u + b~v ) = aT (~u ) + bT (~v )
15/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
For any linear transformation T (~0) = ~0
T (a~u + b~v ) = aT (~u ) + bT (~v )
This has important implications: if you know T (~u ) and T (~v ) ,
then you know the values of T on all the linear combinations of ~u
and ~v .
15/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with these
properties:
For any vectors ~u , ~v ∈ Rn , T (~u + ~v ) = T (~u ) + T (~v )
For any vector ~u ∈ Rn and any c ∈ R, T (c~u ) = cT (~u ).
For any linear transformation T (~0) = ~0
T (a~u + b~v ) = aT (~u ) + bT (~v )
This has important implications: if you know T (~u ) and T (~v ) ,
then you know the values of T on all the linear combinations of ~u
and ~v .
Matrix transformation: Let A be any m × n matrix. Define
T : Rn → Rm by T (~x ) = A~x .
15/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
1
5
0
3
T(
)=
;
T(
)=
0
2
1
4
16/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
1
5
0
3
T(
)=
;
T(
)=
0
2
1
4
−1
What is T (
)?
7
16/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
1
5
0
3
T(
)=
;
T(
)=
0
2
1
4
−1
What is T (
)?
7
−1
1
0
1
0
T(
) = T (−1
+7
) = −1T (
) + 7T (
)
7
0
1
0
1
5
3
16
= −1
+7
=
2
4
26
16/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
1
5
0
3
T(
)=
;
T(
)=
0
2
1
4
−1
What is T (
)?
7
−1
1
0
1
0
T(
) = T (−1
+7
) = −1T (
) + 7T (
)
7
0
1
0
1
5
3
16
= −1
+7
=
2
4
26
In fact,
nothing can stop us from using the same idea to compute
2
T(
) or T (~x ) for any vector ~x ∈ R2 :
−4
Example
16/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
We can carry this much further: All linear transformations
T : Rn → Rm are matrix linear transformations
17/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
We can carry this much further: All linear transformations
T : Rn → Rm are matrix linear transformations
Why?
 
 
 
 
0
0
1
x1
0
1
0
x2 
 
 
 
 
~x =  .  = x1  .  + x2  .  + . . . xn  . 
.
.
.
 .. 
.
.
.
1
0
0
xn
is a linear combination of the vectors
   
 
1
0
0
0 1
0
   
 
 ..  ,  ..  , . . . ,  .. 
. .
.
0
0
1
(standard basis of Rn )
17/24
Prelude
Linear Transformations
Pictorial examples
So by the property of linear transformation
 
 
0
1
1
0
 
 
T (~x ) = x1 T  .  + x2 T  .  + . . . xn T
.
 .. 
 . 
0
0
Matrix Is Everywhere
 
0
0
 
 .. 
 . 
1
only need to know each T (~ej ) where
 
0
 .. 
.
 
th

~ej = 
1 ← j entry
 .. 
.
0
18/24
Prelude
Linear Transformations
Pictorial examples
So by the property of linear transformation
 
 
0
1
1
0
 
 
T (~x ) = x1 T  .  + x2 T  .  + . . . xn T
.
 .. 
 . 
0
0
Matrix Is Everywhere
 
0
0
 
 .. 
 . 
1
only need to know each T (~ej ) where
 
0
 .. 
.
 
th

~ej = 
1 ← j entry
 .. 
.
0
Denote a~j = T (~ej )
T (~x ) = x1 a~1 + x2 a~2 + . . . xn a~n
18/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Ahha: In matrix notation this is written:
T (~x ) = ~a1 ~a2
 
x1

 x2 

. . . ~an  .  = A~x
.
.
xn
19/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Ahha: In matrix notation this is written:
T (~x ) = ~a1 ~a2
 
x1

 x2 

. . . ~an  .  = A~x
.
.
xn
That is, the matrix
A = ~a1 ~a2 . . . ~an
is the matrix of T !
19/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Recap:
20/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) =
20/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) =

x1~a1 + x2~a2 + . . . xn~an =
~a1 ~a2 . . . ~an




x1
x2
..
.



 = A~x .

xn
20/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x ) = T (x1~e1 +x2~e2 +. . . xn~en ) = x1 T (~e1 )+x2 T (~e2 )+. . . xn T (~en ) =

x1~a1 + x2~a2 + . . . xn~an =
~a1 ~a2 . . . ~an




x1
x2
..
.



 = A~x .

xn
Example
20/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear
transformation).
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear
transformation).
Saying that T is 1 to 1 is the same as saying that
~ = ~0 (only trivial solution).
T (~
w ) = ~0 exactly when w
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear
transformation).
Saying that T is 1 to 1 is the same as saying that
~ = ~0 (only trivial solution).
T (~
w ) = ~0 exactly when w
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows.
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear
transformation).
Saying that T is 1 to 1 is the same as saying that
~ = ~0 (only trivial solution).
T (~
w ) = ~0 exactly when w
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows.
The only solution to the homogeneous equation is the zero
solution. And, as a consequence, n ≤ m.
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u ) 6= T (~v ).
Put another words, T (~u ) = T (~v ) implies ~u = ~v .
If T (~u ) = T (~v ) we see that T (~u − ~v ) = ~0 (T linear
transformation).
Saying that T is 1 to 1 is the same as saying that
~ = ~0 (only trivial solution).
T (~
w ) = ~0 exactly when w
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows.
The only solution to the homogeneous equation is the zero
solution. And, as a consequence, n ≤ m.
Examples: shears, contractions and expansions, rotations,
reflections
21/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v .
22/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v .
Saying that T is onto is the same as saying that
T (~u ) = ~v always has a solution.
22/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v .
Saying that T is onto is the same as saying that
T (~u ) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T must
have exactly m non-zero rows.
22/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v .
Saying that T is onto is the same as saying that
T (~u ) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T must
have exactly m non-zero rows.
The non-homogeneous equation must always have a solution. And,
as a consequence, m ≤ n.
22/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm , is there a ~u ∈ Rn such thatT (~u ) = ~v .
Saying that T is onto is the same as saying that
T (~u ) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T must
have exactly m non-zero rows.
The non-homogeneous equation must always have a solution. And,
as a consequence, m ≤ n.
Examples: shears, contractions and expansions, rotations,
reflections
22/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
23/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying that
T is a bijection that respects the vector space structure.
23/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying that
T is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows, the same as the number of columns.
23/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying that
T is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly one
solution.
23/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying that
T is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T must
have exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly one
solution.
Examples: shears, contractions and expansions, rotations,
reflections
23/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearly
independent.
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearly
independent.
T is onto if there is a pivot 1 in every row of the reduced echelon
form.
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearly
independent.
T is onto if there is a pivot 1 in every row of the reduced echelon
form.
Said differently, the column vectors of the matrix of T span the
whole space Rm .
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearly
independent.
T is onto if there is a pivot 1 in every row of the reduced echelon
form.
Said differently, the column vectors of the matrix of T span the
whole space Rm .
T is an isomorphism if there is a pivot 1 in every row and column,
i.e. the reduced echelon matrix is the identity matrix.
24/24
Prelude
Linear Transformations
Pictorial examples
Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) linear
transformation.
T is 1 to 1 if there is a pivot 1 in every column of the reduced
echelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearly
independent.
T is onto if there is a pivot 1 in every row of the reduced echelon
form.
Said differently, the column vectors of the matrix of T span the
whole space Rm .
T is an isomorphism if there is a pivot 1 in every row and column,
i.e. the reduced echelon matrix is the identity matrix.
Said differently, the column vectors of the matrix of T are linearly
independent and span the whole space.
24/24