Mathematics 13: Lecture 15Linear Transformations

Dan Sloughter

Furman University

February 4, 2008

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 1 / 16

Matrix transformations

I Note: If A is an m × n matrix, then we may use A to describe atransformation, or mapping, from Rn to Rm: ~x → A~x .

I That is, we may define a function T : Rn → Rm by T (~x) = A~x .

I Example: If

A =

[1 2 31 −2 −1

]and ~x =

xyz

,then we could define T : R3 → R2 by

T (~x) = A~x =

[1 2 31 −2 −1

]xyz

=

[x + 2y + 3zx − 2y − z

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 2 / 16

Matrix transformations

I Note: If A is an m × n matrix, then we may use A to describe atransformation, or mapping, from Rn to Rm: ~x → A~x .

I That is, we may define a function T : Rn → Rm by T (~x) = A~x .

I Example: If

A =

[1 2 31 −2 −1

]and ~x =

xyz

,then we could define T : R3 → R2 by

T (~x) = A~x =

[1 2 31 −2 −1

]xyz

=

[x + 2y + 3zx − 2y − z

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 2 / 16

Matrix transformations

I Note: If A is an m × n matrix, then we may use A to describe atransformation, or mapping, from Rn to Rm: ~x → A~x .

I That is, we may define a function T : Rn → Rm by T (~x) = A~x .

I Example: If

A =

[1 2 31 −2 −1

]and ~x =

xyz

,then we could define T : R3 → R2 by

T (~x) = A~x =

[1 2 31 −2 −1

]xyz

=

[x + 2y + 3zx − 2y − z

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 2 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation if

I T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),

I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation if

I T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation if

I T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation if

I T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation ifI T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, and

I T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation ifI T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Linearity

I Note: If A is an m × n matrix, ~x and ~y are vectors in Rn, c is ascalar, and T (~x) = A~x , then

I T (~x + ~y) = A(~x + ~y) = A~x + A~y = T (~x) + T (~y),I T (c~x) = A(c~x) = cA~x = cT (~x).

I Definition: We call a function T : Rn → Rm a linear transformation ifI T (~x + ~y) = T (~x) + T (~y) for all ~x and ~y in Rn, andI T (c~x) = cT (~x) for all ~x in Rn and scalars c .

I Theorem: If A is an m × n matrix and T : Rn → Rm is defined byT (~x) = A~x , then T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 3 / 16

Example

I The function T : R2 → R2 defined by

T

([xy

])=

[x2

y2

]is not a linear transformation since, for example,

T

([11

])=

[11

],

but

T

(2

[11

])= T

([22

])=

[44

]6= 2T

([11

]).

I T : R→ R defined by T (x) = 2x + 1 is not a linear transformationsince, for example,

T (1 + 2) = T (3) = 7, but T (1) + T (2) = 3 + 5 = 8.

I However, T (x) = 2x is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 4 / 16

Example

I The function T : R2 → R2 defined by

T

([xy

])=

[x2

y2

]is not a linear transformation since, for example,

T

([11

])=

[11

],

but

T

(2

[11

])= T

([22

])=

[44

]6= 2T

([11

]).

I T : R→ R defined by T (x) = 2x + 1 is not a linear transformationsince, for example,

T (1 + 2) = T (3) = 7, but T (1) + T (2) = 3 + 5 = 8.

I However, T (x) = 2x is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 4 / 16

Example

I The function T : R2 → R2 defined by

T

([xy

])=

[x2

y2

]is not a linear transformation since, for example,

T

([11

])=

[11

],

but

T

(2

[11

])= T

([22

])=

[44

]6= 2T

([11

]).

I T : R→ R defined by T (x) = 2x + 1 is not a linear transformationsince, for example,

T (1 + 2) = T (3) = 7, but T (1) + T (2) = 3 + 5 = 8.

I However, T (x) = 2x is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 4 / 16

Theorem

I If T : Rn → Rm is a linear transformation, then there exists an m × nmatrix A for which T (~x) = A~x for any vector ~x in Rn.

I Reason:

I Let A be the matrix with columns T (~e1), T (~e2), . . . , T (~en).I Let ~x be a vector in Rn with

~x =

x1

x2

...xn

= x1~e1 + ~e2 + · · ·+ xn~en.

I ThenT (~x) = x1T (~e1) + x2T (~e2) + · · ·+ xnT (~en) = A~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 5 / 16

Theorem

I If T : Rn → Rm is a linear transformation, then there exists an m × nmatrix A for which T (~x) = A~x for any vector ~x in Rn.

I Reason:

I Let A be the matrix with columns T (~e1), T (~e2), . . . , T (~en).I Let ~x be a vector in Rn with

~x =

x1

x2

...xn

= x1~e1 + ~e2 + · · ·+ xn~en.

I ThenT (~x) = x1T (~e1) + x2T (~e2) + · · ·+ xnT (~en) = A~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 5 / 16

Theorem

I If T : Rn → Rm is a linear transformation, then there exists an m × nmatrix A for which T (~x) = A~x for any vector ~x in Rn.

I Reason:I Let A be the matrix with columns T (~e1), T (~e2), . . . , T (~en).

I Let ~x be a vector in Rn with

~x =

x1

x2

...xn

= x1~e1 + ~e2 + · · ·+ xn~en.

I ThenT (~x) = x1T (~e1) + x2T (~e2) + · · ·+ xnT (~en) = A~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 5 / 16

Theorem

I If T : Rn → Rm is a linear transformation, then there exists an m × nmatrix A for which T (~x) = A~x for any vector ~x in Rn.

I Reason:I Let A be the matrix with columns T (~e1), T (~e2), . . . , T (~en).I Let ~x be a vector in Rn with

~x =

x1

x2

...xn

= x1~e1 + ~e2 + · · ·+ xn~en.

I ThenT (~x) = x1T (~e1) + x2T (~e2) + · · ·+ xnT (~en) = A~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 5 / 16

Theorem

I If T : Rn → Rm is a linear transformation, then there exists an m × nmatrix A for which T (~x) = A~x for any vector ~x in Rn.

I Reason:I Let A be the matrix with columns T (~e1), T (~e2), . . . , T (~en).I Let ~x be a vector in Rn with

~x =

x1

x2

...xn

= x1~e1 + ~e2 + · · ·+ xn~en.

I ThenT (~x) = x1T (~e1) + x2T (~e2) + · · ·+ xnT (~en) = A~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 5 / 16

Example

I Suppose T : R3 → R2 is defined by

T

xyz

=

[x − y + 3z

3x − 4y + 5z

].

I It follows that

T

xyz

=

[1 −1 33 −4 5

]xyz

.I So T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 6 / 16

Example

I Suppose T : R3 → R2 is defined by

T

xyz

=

[x − y + 3z

3x − 4y + 5z

].

I It follows that

T

xyz

=

[1 −1 33 −4 5

]xyz

.

I So T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 6 / 16

Example

I Suppose T : R3 → R2 is defined by

T

xyz

=

[x − y + 3z

3x − 4y + 5z

].

I It follows that

T

xyz

=

[1 −1 33 −4 5

]xyz

.I So T is a linear transformation.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 6 / 16

Example

I Let Rθ : R2 → R2 be the transformation which rotates a vector in R2

through an angle θ.

I Note:

Rθ(~e1) =

[cos(θ)sin(θ)

]and Rθ(~e2) =

[cos(θ + π

2

)sin(θ + π

2

)] =

[− sin(θ)

cos(θ)

].

I It follows that, if Rθ is linear, then Rθ(~v) = A~v where

A =

[cos(θ) − sin(θ)sin(θ) cos(θ)

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 7 / 16

Example

I Let Rθ : R2 → R2 be the transformation which rotates a vector in R2

through an angle θ.

I Note:

Rθ(~e1) =

[cos(θ)sin(θ)

]and Rθ(~e2) =

[cos(θ + π

2

)sin(θ + π

2

)] =

[− sin(θ)

cos(θ)

].

I It follows that, if Rθ is linear, then Rθ(~v) = A~v where

A =

[cos(θ) − sin(θ)sin(θ) cos(θ)

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 7 / 16

Example

I Let Rθ : R2 → R2 be the transformation which rotates a vector in R2

through an angle θ.

I Note:

Rθ(~e1) =

[cos(θ)sin(θ)

]and Rθ(~e2) =

[cos(θ + π

2

)sin(θ + π

2

)] =

[− sin(θ)

cos(θ)

].

I It follows that, if Rθ is linear, then Rθ(~v) = A~v where

A =

[cos(θ) − sin(θ)sin(θ) cos(θ)

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 7 / 16

Example (cont’d)

I Now for any ~v =

[xy

], we have

A~v =

[x cos(θ)− y sin(θ)x sin(θ) + y cos(θ)

].

I It is easy to see than that A~v · A~v = x2 + y2 andA~v · ~v = (x2 + y2) cos(θ).

I Hence ‖A~v‖ = ‖~v‖ and the angle between ~v and A~v is

A~v · ~v‖A~v‖‖~v‖

= cos(θ).

I That is, Rθ(~v) = A~v .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 8 / 16

Example (cont’d)

I Now for any ~v =

[xy

], we have

A~v =

[x cos(θ)− y sin(θ)x sin(θ) + y cos(θ)

].

I It is easy to see than that A~v · A~v = x2 + y2 andA~v · ~v = (x2 + y2) cos(θ).

I Hence ‖A~v‖ = ‖~v‖ and the angle between ~v and A~v is

A~v · ~v‖A~v‖‖~v‖

= cos(θ).

I That is, Rθ(~v) = A~v .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 8 / 16

Example (cont’d)

I Now for any ~v =

[xy

], we have

A~v =

[x cos(θ)− y sin(θ)x sin(θ) + y cos(θ)

].

I It is easy to see than that A~v · A~v = x2 + y2 andA~v · ~v = (x2 + y2) cos(θ).

I Hence ‖A~v‖ = ‖~v‖ and the angle between ~v and A~v is

A~v · ~v‖A~v‖‖~v‖

= cos(θ).

I That is, Rθ(~v) = A~v .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 8 / 16

Example (cont’d)

I Now for any ~v =

[xy

], we have

A~v =

[x cos(θ)− y sin(θ)x sin(θ) + y cos(θ)

].

I It is easy to see than that A~v · A~v = x2 + y2 andA~v · ~v = (x2 + y2) cos(θ).

I Hence ‖A~v‖ = ‖~v‖ and the angle between ~v and A~v is

A~v · ~v‖A~v‖‖~v‖

= cos(θ).

I That is, Rθ(~v) = A~v .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 8 / 16

Compositions

I Notation: If T : Rn → Rm with T (~x) = Ax for an m × n matrix A,we write [T ] = A.

I If T : Rn → Rm and S : Rm → Rp, then [S ◦ T ] = [S ][T ].

I Reason: For any vector ~x in Rm,

S ◦ T (~x) = S(T (~x)) = S([T ]~x) = [S ][T ]~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 9 / 16

Compositions

I Notation: If T : Rn → Rm with T (~x) = Ax for an m × n matrix A,we write [T ] = A.

I If T : Rn → Rm and S : Rm → Rp, then [S ◦ T ] = [S ][T ].

I Reason: For any vector ~x in Rm,

S ◦ T (~x) = S(T (~x)) = S([T ]~x) = [S ][T ]~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 9 / 16

Compositions

I Notation: If T : Rn → Rm with T (~x) = Ax for an m × n matrix A,we write [T ] = A.

I If T : Rn → Rm and S : Rm → Rp, then [S ◦ T ] = [S ][T ].

I Reason: For any vector ~x in Rm,

S ◦ T (~x) = S(T (~x)) = S([T ]~x) = [S ][T ]~x .

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 9 / 16

Example

I Suppose T : R2 → R3 is defined by

T

([xy

])=

x + yx − y

3x

and S : R3 → R2 is defined by

S

xyz

=

[x − y + z

x + y

].

I Then

[T ] =

1 11 −13 0

and [S ] =

[1 −1 11 1 0

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 10 / 16

Example

I Suppose T : R2 → R3 is defined by

T

([xy

])=

x + yx − y

3x

and S : R3 → R2 is defined by

S

xyz

=

[x − y + z

x + y

].

I Then

[T ] =

1 11 −13 0

and [S ] =

[1 −1 11 1 0

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 10 / 16

Example (cont’d)

I Then S ◦ T : R2 → R2 and

[S ◦ T ] =

[1 −1 11 1 0

]1 11 −13 0

=

[3 22 0

].

I That is,

S ◦ T

([xy

])=

[3 22 0

] [xy

]=

[3x + 2y

2x

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 11 / 16

Example (cont’d)

I Then S ◦ T : R2 → R2 and

[S ◦ T ] =

[1 −1 11 1 0

]1 11 −13 0

=

[3 22 0

].

I That is,

S ◦ T

([xy

])=

[3 22 0

] [xy

]=

[3x + 2y

2x

].

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 11 / 16

Example

I Suppose S reflects a vector in R2 about the x-axis and T rotates avector in R2 through an angle π.

I Then

S

([xy

])=

[1 00 −1

] [xy

]and T

([xy

])=

[−1 0

0 −1

] [xy

].

I So

(S ◦ T )

([xy

])=

[1 00 −1

] [−1 0

0 −1

] [xy

]=

[−1 0

0 1

] [xy

].

I Hence S ◦ T is a reflection about the y -axis.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 12 / 16

Example

I Suppose S reflects a vector in R2 about the x-axis and T rotates avector in R2 through an angle π.

I Then

S

([xy

])=

[1 00 −1

] [xy

]and T

([xy

])=

[−1 0

0 −1

] [xy

].

I So

(S ◦ T )

([xy

])=

[1 00 −1

] [−1 0

0 −1

] [xy

]=

[−1 0

0 1

] [xy

].

I Hence S ◦ T is a reflection about the y -axis.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 12 / 16

Example

I Suppose S reflects a vector in R2 about the x-axis and T rotates avector in R2 through an angle π.

I Then

S

([xy

])=

[1 00 −1

] [xy

]and T

([xy

])=

[−1 0

0 −1

] [xy

].

I So

(S ◦ T )

([xy

])=

[1 00 −1

] [−1 0

0 −1

] [xy

]=

[−1 0

0 1

] [xy

].

I Hence S ◦ T is a reflection about the y -axis.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 12 / 16

Example

I Suppose S reflects a vector in R2 about the x-axis and T rotates avector in R2 through an angle π.

I Then

S

([xy

])=

[1 00 −1

] [xy

]and T

([xy

])=

[−1 0

0 −1

] [xy

].

I So

(S ◦ T )

([xy

])=

[1 00 −1

] [−1 0

0 −1

] [xy

]=

[−1 0

0 1

] [xy

].

I Hence S ◦ T is a reflection about the y -axis.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 12 / 16

Inverses

I We say a linear transformation T : Rn → Rn is invertible if thereexists a linear transformation S : Rn → Rn such that, for any ~x in Rn,

(S ◦ T )(~x) = ~x and (T ◦ S)(~x) = ~x .

I We let T−1 denote the inverse of T .

I Note: if A = [T ] and B = [T−1], then we must have BA = I andAB = I .

I That is, [T−1] = A−1.

I In particular, T is invertible if and only if [T ] is invertible.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 13 / 16

Inverses

I We say a linear transformation T : Rn → Rn is invertible if thereexists a linear transformation S : Rn → Rn such that, for any ~x in Rn,

(S ◦ T )(~x) = ~x and (T ◦ S)(~x) = ~x .

I We let T−1 denote the inverse of T .

I Note: if A = [T ] and B = [T−1], then we must have BA = I andAB = I .

I That is, [T−1] = A−1.

I In particular, T is invertible if and only if [T ] is invertible.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 13 / 16

Inverses

I We say a linear transformation T : Rn → Rn is invertible if thereexists a linear transformation S : Rn → Rn such that, for any ~x in Rn,

(S ◦ T )(~x) = ~x and (T ◦ S)(~x) = ~x .

I We let T−1 denote the inverse of T .

I Note: if A = [T ] and B = [T−1], then we must have BA = I andAB = I .

I That is, [T−1] = A−1.

I In particular, T is invertible if and only if [T ] is invertible.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 13 / 16

Inverses

I We say a linear transformation T : Rn → Rn is invertible if thereexists a linear transformation S : Rn → Rn such that, for any ~x in Rn,

(S ◦ T )(~x) = ~x and (T ◦ S)(~x) = ~x .

I We let T−1 denote the inverse of T .

I Note: if A = [T ] and B = [T−1], then we must have BA = I andAB = I .

I That is, [T−1] = A−1.

I In particular, T is invertible if and only if [T ] is invertible.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 13 / 16

Inverses

I We say a linear transformation T : Rn → Rn is invertible if thereexists a linear transformation S : Rn → Rn such that, for any ~x in Rn,

(S ◦ T )(~x) = ~x and (T ◦ S)(~x) = ~x .

I We let T−1 denote the inverse of T .

I Note: if A = [T ] and B = [T−1], then we must have BA = I andAB = I .

I That is, [T−1] = A−1.

I In particular, T is invertible if and only if [T ] is invertible.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 13 / 16

Example

I Let Rθ : R2 → R2 be the linear transformation which rotates a vectorthrough an angle θ.

I Clearly, we should have R−1θ = R−θ.

I That is,

[Rθ] =

[cos(θ) − sin(θ)sin(θ) cos(θ)

],

and

[R−1θ ] = [Rθ]−1 =

[cos(−θ) − sin(−θ)sin(−θ) cos(−θ)

]=

[cos(θ) sin(θ)− sin(θ) cos(θ)

],

which may be verified easily.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 14 / 16

Example

I Let Rθ : R2 → R2 be the linear transformation which rotates a vectorthrough an angle θ.

I Clearly, we should have R−1θ = R−θ.

I That is,

[Rθ] =

[cos(θ) − sin(θ)sin(θ) cos(θ)

],

and

[R−1θ ] = [Rθ]−1 =

[cos(−θ) − sin(−θ)sin(−θ) cos(−θ)

]=

[cos(θ) sin(θ)− sin(θ) cos(θ)

],

which may be verified easily.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 14 / 16

Example

I Let Rθ : R2 → R2 be the linear transformation which rotates a vectorthrough an angle θ.

I Clearly, we should have R−1θ = R−θ.

I That is,

[Rθ] =

[cos(θ) − sin(θ)sin(θ) cos(θ)

],

and

[R−1θ ] = [Rθ]−1 =

[cos(−θ) − sin(−θ)sin(−θ) cos(−θ)

]=

[cos(θ) sin(θ)− sin(θ) cos(θ)

],

which may be verified easily.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 14 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,I S which reflects a vector about the x-axis, andI U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,I S which reflects a vector about the x-axis, andI U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,I S which reflects a vector about the x-axis, andI U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,

I S which reflects a vector about the x-axis, andI U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,I S which reflects a vector about the x-axis, and

I U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example

I Suppose ` is the line through the origin in R2 which makes an angle θwith the x-axis.

I Let T : R2 → R2 be the transformation which reflects a vector ~xabout `.

I We make think of T as the composition of three lineartransformations:

I P which rotates a vector by the angle −θ,I S which reflects a vector about the x-axis, andI U which rotates a vector by the angle θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 15 / 16

Example (cont’d)

I Hence the matrix for T should be[cos(θ) − sin(θ)sin(θ) cos(θ)

] [1 00 −1

] [cos(θ) sin(θ)− sin(θ) cos(θ)

]=

[cos(θ) − sin(θ)sin(θ) cos(θ)

] [cos(θ) sin(θ)sin(θ) − cos(θ)

]=

[cos(2θ) sin(2θ)sin(2θ) − cos(2θ)

].

I Note: [cos(2θ) sin(2θ)sin(2θ) − cos(2θ)

]=

[cos(2θ) − sin(2θ)sin(2θ) cos(2θ)

] [1 00 −1

],

so a reflection about ` is the composition of a reflection about thex-axis with a rotation through an angle 2θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 16 / 16

Example (cont’d)

I Hence the matrix for T should be[cos(θ) − sin(θ)sin(θ) cos(θ)

] [1 00 −1

] [cos(θ) sin(θ)− sin(θ) cos(θ)

]=

[cos(θ) − sin(θ)sin(θ) cos(θ)

] [cos(θ) sin(θ)sin(θ) − cos(θ)

]=

[cos(2θ) sin(2θ)sin(2θ) − cos(2θ)

].

I Note: [cos(2θ) sin(2θ)sin(2θ) − cos(2θ)

]=

[cos(2θ) − sin(2θ)sin(2θ) cos(2θ)

] [1 00 −1

],

so a reflection about ` is the composition of a reflection about thex-axis with a rotation through an angle 2θ.

Dan Sloughter (Furman University) Mathematics 13: Lecture 15 February 4, 2008 16 / 16