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Chapter 2 – Linear Transformations

Outline2.1 Introduction to Linear Transformations and Their Inverses2.2 Linear Transformations in Geometry2.3 The Inverse of a Linear Transformation2.4 Matrix Products

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2.1 Introduction to Linear Transformations and Their Inverses

•

• The matrix A is called the coefficient matrix of the transformation.

• is called a linear transformation.• : the decoding transformation.• : the inverse of the coding

transformation.• Since the decoding transformation is

the inverse of the coding transformation , we say that the matrix B is the inverse of the matrix A. We can write this as .

xAy

x

x

xx

xx

y

y

2

1

21

21

2

1

52

31

52

3xAy

xAy

xy

yx

yBx

xAy

1AB

3

Invertible

• No all linear transformations are invertible.

• Suppose with matrix , the solutions are

– Because this system does not have a unique solution, it is impossible to recover the actual position from the encoded position. The coding matrix A is noninvertible.

• Consider two sets X and Y. A function T from X to Y is a rule that associates with each element x of X a unique element y of Y.– The set X is called the domain of the function, and Y is its codomain.

We will sometimes refer to x as the input of the function and to y as its output.

212

211

42

2

xxy

xxy

42

21A

t

t

x

x 289

2

1

4

Linear Transformations

• (Definition 2.1.1) A function T from Rn to Rm is called a linear transformation if there is an m ×n matrix A such that for all in Rn.

• A linear transformation is a special kind of function.

• The identity transformation from Rn to Rn: all entries on the main diagonal are 1, and all other entries are 0. This matrix is called the identity matrix and is denoted by In.

• , where . The output vector is obtained from by rotating through an angle of 90o in the counterclockwise.

xAxT

)(

x

xAy

01

10A y

x

5

(Fact 2.1.2) The Column of the Matrix of a Linear Transformation

• (Fact 2.1.2) Consider a linear transformation T from Rm to Rn. Then, the matrix of T is

– To justify this result, we have

then

– The vectors in Rm are sometimes referred to as the standard vectors in Rm. The standard vectors in R3 are often denoted by

|||

|||

21 mvvvA

imiii vvvvveAeT

0

1

0

0

||||

||||

21

meee

,, 21

321 ,, eee

kji

,,

0

1

0

0

where,

|||

)()()(

|||

21

im eeTeTeTA

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(Fact 2.1.3) Linear Transformations

• (Fact 2,1,3) A transformation T from Rn to Rm is linear if (and only if)– , for all ,

in Rn, and

– , for all in Rn and all scalars k.

wTvTwvT

v

w

vkTvkT

v

7

2.2 Linear Transformations in Geometry

• (Example 1) Consider the matrices

Show the effect of each of these matrices on our standard letter L, and describe each transformation in words.

10

01,

00

01,

20

02CBA

11

11,

10

5.01,

01

10FED

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Scalings

• For any positive constant k, the matrix defines a scaling by k, since

This is a dilation (or enlargement) if k exceeds 1, and it is a contraction (or shrinking) for values of k between 0 and 1.

k

k

0

0

xkx

xk

kx

kx

x

x

k

kx

k

k

2

1

2

1

2

1

0

0

0

0

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Projections

• Consider a line L in the plane, running through the origin. Any vector in R2 can be written uniquely as

where is parallel to line L, and is perpendicular to L.

• The transformation from R2 to R2 is called the projection of onto L, often denoted by :

x

xxx ||

||x x

||xxT

x xL

proj

||proj xxL

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Example 2

• (Example 2) Find the matrix A of the projection onto the line L spanned by

3

4v

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Definition 2.2.1 Projections

• (Definition 2.2.1) Consider a line L in the coordinate plane, running through the origin. Any vector in R2 can be written uniquely as

where is parallel to line L, and is perpendicular to L.

• The transformation from R2 to R2 is called the projection of onto L, often denoted by . If is a unit vector parallel to L, then

The transformation is linear, with matrix

xxx ||

x

||x x

||xxT

x xL

proj

2

1

u

uu

.)()(proj uuxxL

)(proj)( xxT L

2221

2121

uuu

uuu

12

Reflections

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Definition 2.2.2 Reflections

• (Definition 2.2.2) Consider a line L in the coordinate plane, running through the origin, and let be a vector in R2. The linear transformation is called the reflection of about L, often denoted by :

We have a formula relating to :

The matrix of T is of the form , where a2+b2=1.

xxx ||

xxxT ||)( x

)(ref xL

xxxL

||)(ref

)(ref xL

xL

proj

.)(2)(proj2)(ref xuuxxxx LL

ab

ba

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Projections and Reflections in Space

• Let be the plane through the origin perpendicular to L; note that the vector will be parallel to . We can give formulas for the orthogonal projection onto V, as well as for the reflections about V and L, in terms of the projection onto L:

VL

VL x

uuxxxxx LV

)()(proj)(proj

xuuxxxxxx LVLL

)(2)(proj2)(proj)(proj)(ref

uuxxxxxx LLVV

)(2)(ref)(proj)(proj)(ref

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Example 3

• (Example 3) Let V be the plane defined by 2x1+x2-2x3=0, and let . Find

2

4

5

x

)(ref xV

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Fact 2.2.3 Rotations

• (Fact 2.2.3) The matrix of a counterclockwise rotation in R2 through an angle θ is

Note that this matrix is of the form , where a2+b2=1.

cossin

sincos

ab

ba

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Rotations

• The matrix of a counterclockwise rotation through an angle is

• The matrix of a counterclockwise rotation through an angle of is

cossin

sincos

01

10

cossin

sincos

22

22

2

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Example 4 &5

• (Example 4) The matrix of a counterclockwise rotation through π/6 (or 30o) is

• (Example 5) Examine how the linear transformation

affects our standard letter L. Here a and b are arbitrary constants.

.31

13

2

1

)6/cos()6/sin(

)6/sin()6/cos(

xab

baxT

)(

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Fact 2.2.4 Rotations Combined with a Scaling

• (Fact 2.2.4) A matrix of the form represents a rotation combined with a scaling.More precisely, if r and θ are the polar coordinates of

vector , then represents a rotation through θ combined with a scaling by r.

ab

ba

b

a

ab

ba

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Rotation-Dilations

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Shears

• Let L be a line in R2. A linear transformation T from R2 to R2 is called a shear parallel to L if– , for all vector on L,

and

– is parallel to L for all vectors in R2.

vvT

v

xxT

x

xy

10

1 21

22

Fact 2.2.5 Horizontal and Vertical Shears

• (Fact 2.2.5) The matrix of a horizontal shear is of the

form , and the matrix of a vertical shear is of the

form , where k is an arbitrary constant.

10

1 k

1

01

k

23

Shears (II)

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2.3 The Inverse of Linear Transformation

• (Definition 2.3.1) A function T from X to Y is called invertible if the equation T(x)=y has a unique solution x in X for each y in Y.

• If a function T is invertible, then so is T-1, and (T-1)-1=T.

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Invertible

• Consider the linear transformation from Rn to Rm given bywhere A is an matrix– m<n

The system has either no solutions or infinitely many solutions. The transformation is noninvertible.

– m=nThe system has a unique solution if and only if rref(A)=In. Therefore, the transformation is invertible if and only if reff(A)=In, or equivalently, if rank(A)=n.

– m>nThe transformation is noninvertible since the system is inconsistent.

nmxAy

xAy

xAy

xAy

x

xAy

xAy

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Invertible Matrices

• (Definition 2.3.2) A matrix A is called invertible if the linear transformation is invertible. The matrix of the inverse transformation is denoted by A-1. If the transformation is invertible, its inverse is .

• (Fact 2.3.3) An matrix A is invertible if and only if– A is a square matrix (i.e., m=n), and

– rref(A)=In.

• (Fact 2.3.4) Let A be an matrix– Consider a vector in Rn. If A is invertible, then the system has

the unique solution . If A is noninvertible, then the systemhas infinitely many solutions or none.

– Consider the special case when . The system has as a solution. If A is invertible, then this is the only solution. If A is noninvertible, then there are infinitely many other solutions.

xAy

xAy

yAx 1

nm

nmb

bxA

bAx 1 bxA

0

b 0

xA 0

x

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Example 1

• (Example 1) Is the matrix A invertible?

987

654

321

A

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Finding the Inverse of a Matrix

• If a matrix A is invertible, how can we find the inverse matrix A-1?

283

232

111

A

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Finding the Inverse of a Matrix

• (Fact 2.3.5) To find the inverse of an n×n matrix A, form the n× (2n) matrix [A|In] and compute rref[A|In].– If rref[A|In] is of the form [In|B], then A is invertible, and A-1=B.

– If rref[A|In] is of another form (i.e., its left half fails to be In), then A is not invertible. Note that the left half of rref[A|In] is rref(A).

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Inverse and Determinant of a 2×2 Matrix

• The 2×2 matrix

is invertible if (and only if) ad-bc≠0.Quantity ad-bc is called the determinant of A, written det(A):

• If

is invertible, then

dc

baA

.det)det( bcaddc

baA

dc

baA

.)det(

111

ac

bd

Aac

bd

bcaddc

ba

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2.4 Matrix Products

• The composite of two functions: The composite of the functions y=sin(x) and z=cos(y) is z=cos(sin(x)).

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Matrix Multiplication

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Definition 2.4.1 Matrix Multiplication

• (Definition 2.4.1)– Let B be an n×p matrix and A a q×m matrix. The product BA is

defined if (and only if) p=q.

– If B is an n×p matrix and A a p×m matrix, then the product BA is defined as the matrix of the linear transformation . This means that , for all in Rm. The product BA is an n×m matrix.

)()( xABxT

xBAxABxT

)()()( x

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Fact 2.4.2 The Columns of the Matrix Product

• (Fact 2.4.2) Let B be n×p matrix and A a p×m matrix with columns Then, the product BA is

To find BA, we can multiply B with the columns of A and combine the resulting vectors.

.,,, 21 mvvv

|||

|||

|||

|||

2121 mm vBvBvBvvvBBA

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Fact 2.4.3

• (Fact 2.4.3)– Matrix multiplication is noncommutative.

– AB≠BA, in general. However, at times it does happen that AB=BA; then we say that the matrices A and B commute.

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Fact 2.4.4 The Entries of the Matrix Product

• (Fact 2.4.4) Let B be an n×p matrix and A a p×m matrix. The ijth entry of BA is the dot product of the ith row of B with the jth column of A.

Is the n×m matrix whose ijth entry is

pmpjpp

mj

mj

npnn

ipii

p

p

aaaa

aaaa

aaaa

bbb

bbb

bbb

bbb

BA

21

222221

111211

21

21

22221

11211

p

kkjikpjipjiji abababab

12211

37

Example 1

53

21

98

76• (Example 1)

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Fact 2.4.5 Multiplying with the Inverse

• (Fact 2.4.5) For an invertible n×n matrix A,

. and 11nn IAAIAA

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Fact 2.4.6 Multiplying with the Identity Matrix

• (Fact 2.4.6) For an n×m matrix A,.AAIAI nm

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Fact 2.4.7 Matrix Multiplication is Associative

• (Fact 2.4.7) Matrix multiplication is associative (AB)C=A(BC)We can simply write ABC for the product (AB)C=A(BC).

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Fact 2.4.8 The Inverse of a Product of Matrices

• (Fact 2.4.8) If A and B are invertible n×n matrices, then BA is invertible as well, and (BA)-1=A-1B-1.Pay attention to the order of the matrices. (Order matters!)

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Fact 2.4.9 A Criterion for Invertibility

• (Fact 2.4.9) Let A and B be two n×n matrices such that BA=In

Then,– A and B are both invertible,

– A-1=B and B-1=A, and

– AB=In.

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Example 2

• (Example 2) Suppose A, B, and C are three n×n matrices and ABC=In. Show that B is invertible, and express B-1 in terms of A and C.

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Fact 2.4.10 Distributive Property for Matrices

• (Fact 2.4.10) If A and B are n×p matrices, and C and D are p×m matrices, then A(C+D)=AC+AD, and (A+B)C=AC+BC.

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Fact 2.4.11

• (Fact 2.4.11) If A is an n×p matrix, B is a p×m matrix, and k is a scaler, then (kA)B=A(kB)=k(AB).

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Fact 2.4.12 Multiplying Partitioned Matrices

• (Fact 2.4.12) Partitioned matrices can be multiplied as though the submatrices were scalars (i.e., using the formula in Fact 2.4.4):

is the partitioned matrix whose ijth “entry” is the matrix

provided that all the products AikBkj are defined.

pmpjpp

mj

mj

npnn

ipii

p

p

BBBB

BBBB

BBBB

AAA

AAA

AAA

AAA

AB

21

222221

111211

21

21

22221

11211

p

kkjikpjipjiji BABABABA

12211

47

Example 3

• (Example 3)

987

654

321

101

110

48

Example 4

• (Example 4) Let A be a partitioned matrix

where A11 is an n×n matrix, A22 is an m×m matrix, and A12 is an n×m matrix.– For which choices of A11, A12, and A22 is A invertible?

– If A is invertible, what is A-1 (in terms of A11, A12, A22)?

22

1211

0 A

AAA

49

Example 5

• (Example 5) 1

10000

01000

00100

65421

32111