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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
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