LA Ch 6 Lab - SGLee

Chapter 6

Linear Transformations

6.1 Matrix as a Function (Transformation)

6.2 Geometric Meaning of Linear Transformations

6.3 Kernel and Range

6.4 Composition of Linear Transformations and Invertibility

6.5*Computer Graphics with Sage

Exercises

So far, we have considered matrix mainly as a coefficient matrix from systems of linear equations. Now, we consider amatrix as a function.

We have observed that the set of vectors together with two operations reborn as an algebraic structure, namely asubspace(vector space).

Matrix will be reborn as a linear transformation, which is a function that preserves the operations in a vector space.

And linear transformations are used for noise filtering in signal processing and analysis in engineering processes.

We show a linear transformation from -dimensional space  to -dimensional space  can be expressed as a matrix .

We shall also look at geometric meaning of linear transformations from  to  and applications in computer graphics.

Textbook download:  (New English and Korean Version)

6.1 Matrix as a Function (Transformation)

Reference video: https://youtu.be/KHTR69HDSNs http://youtu.be/YF6-ENHfI6E

Matrix can be considered as a special function with the property of linearity.

Such a function play an important role in science and various areas in daily life,

such as mathematics, physics, engineering control theory, image processing, sound signal, and computer graphics.

What is a Transformation?

 Definition [Transformation] A function, whose input and output are both vectors, is called a transformation. For a transformation ,  is called an image of  by , and  is called a pre-image of .           As a special case of transformations, , for some  matrix  and a vector ,   is called a matrix transformation.

 [Remark] Computer simulation [Matrix transformation)

 Definition [Linear Transformation] If a transformation  from  to , satisfies the following two conditions for any vectors  and for any scalar ,       (1)                (2)        then  is called a linear transformation from  to .  Especially, a linear transformation from  to  itself,                     is called a linear operator on .

Show that  is a linear transformation if we define , for any vector  in , as follows

For any two vectors  in  and for any scalar ,

(1)

.

(2) .

Therefore, by definition,  is a linear transformation from  to .

Let. Show that T is a linear transformation.

For any two vectors  and  in  and for any scalar ,

(1)

.

(2)

Therefore, T is a linear transformation.

This type of linear transformation is called orthogonal projection on -plane.

If we define  as follows, show that  is not a linear transformation.

.

For any two vectors,  in ,

.

However,

. Hence .

Therefore, we conclude that  is not a linear transformation.

 [Remark] Special Linear Transformations zero transformation: For any , if we define  as ,       then  is a linear transformation. This is called a zero transformation.    identity transformation: For any , if we define  as ,      then  is a linear transformation. This is called an identity transformation.     matrix transformation: For any  matrix  and for any vector  in ,       if we define ,      then  is a linear transformation from  to . This is called a matrix transformation.

Let is defined as follows. Show  is a linear transformation.

For any two vectors  in  and for any scalar ,

(1)

(2)

and hence,  is a linear transformation.

A linear transformation  from  is

,

is a matrix transformation for a matrix .

 Theorem 6.1.1[Properties of Linear Transformation] If  is a linear transformation, then it satisfies the following conditions:    (1) .  (2) .  (3) .

(1) Since .

(2)

(3)

Each linear transformation  from  to  can be expressed as a matrix transformation.

Let  be any linear transformation. For elementary unit vectors,

of  and for any , we have

and   as  ,  are  matrix, we can write them as

.

Therefore any linear transformation  can be expressed as

(1)

Now let  be an  matrix which has  ,  as it's columns.

Then,

.

The above matrix  is called the standard matrix of  and is denoted by .

Hence, the standard matrix of the linear transformation given by (1) can be found easily from

the column vectors by substituting the elementary unit vectors to  in that order.

 Theorem  6.1.2[Standard Matrix] If  is a linear transformation,     then the standard matrix   of  T  has the following relation for .                                           ,       where .

For a linear transformation ,

by using the standard matrix of , rewrite it as .

Let , which columns are , then

as .

[ 1  2  0]

[-1 -1  0]

[ 0  0  1]

[ 1  0  1]

Vector space of dimension 3 over Rational Field

Vector space of dimension 4 over Rational Field

(-4, 1, 3, 5)

(-4, 1, 3, 5)                                                                     ■

6.2 Geometric Meaning of Linear Transformations

Reference video: https://youtu.be/hJXsKubtgms https://youtu.be/cgySDj-OVlM

In this section, we study the geometric meaning of linear transformations.

For a given image, continuous showing of series of images with small variations makes a motion picture.

Linear transformation can be applied to computer graphics and numerical algorithms, and it is an essential tool for many areas such as animation.

Linear Transformation from  to

A linear transformation  defined by  moves a vector

to an another vector  .

[rotation, symmetry, orthogonal projection]

We illustrate a few linear transformations on .

(1)  is a linear transformation which rotates a vector in  counterclockwise by  around the origin.

(2) An orthogonal projection  on -axis is a linear

transformation.

(3) A symmetric movement  around -axis is a linear transformation.

Find the standard matrix  for a linear transformation which moves a point  in

to a symmetric image around the given line.

(1) -axis                      (2) line

Symmetric (linear) transformation around -axis and the line  are given in the following figures.

,

,      .

 [Remark]  Simulation [linear transformation] [symmetric transformations and orthogonal projection transformations]            [rotation]

Linear transformation  which moves any vector  in  to a symmetric image around a line,

which passes through the origin with angle  between the -axis and the line,

can be expressed by the following matrix presentation .

.

In , if , i.e. .

As shown in the picture, let us define an orthogonal projection as a linear transformation (linear operator)

which maps any vector  in  to the orthogonal projection on a line,

which passes through the origin with angle  between the -axis and the line.

Let us denote the standard matrix correspond to  is .

(the same direction with a half length)

In , if  is a projection onto the -axis.

 [Remark] shear transformations (computer simulation) (1) : shear transformation along the -axis with scale  (2) : shear transformation along the -axis with scale

 Definition   [Isometry] A linear transformation , which preserve the magnitude  (or length of a vector), , is called Euclidean isometry.

 Theorem  6.2.1 For a linear operator , the following statements are equivalent:    (1) ,            [isometry].  (2) ,   [preserve the dot (inner) product].

 Definition  [Orthogonal matrix] For a square matrix , if  then  is called an orthogonal matrix.

For any real number  is an orthogonal matrix, and  .

Verify the following matrices are orthogonal matrix.

Verify  by using the Sage.

[1 0]       [1 0 0]

[0 1]       [0 1 0]

[0 0 1]                                                                                                    ■

 Theorem6.2.2 For any  matrix , the following statements hold:    (1) The transpose of an orthogonal matrix is an orthogonal matrix.  (2) The inverse of an orthogonal matrix is an orthogonal matrix.  (3) The product of orthogonal matrices is an orthogonal matrix.  (4) If    is an orthogonal matrix, then  or .

(1) and (2) are left as an exercise to the reader.

(3) If  and , then

and hence  is an orthogonal matrix.

(4) Observe that

or  .

 Theorem  6.2.3 For any  matrix , the following statements are equivalent:    (1)  is an orthogonal matrix.  (2) , .  (3) , .  (4) The row vectors of  are orthonormal.  (5) The column vectors of  are orthonormal.

(1)  (2):

(2)  (3):

and

.

Hence  .

(3)  (1):

(Kronecker's delta)

We skip the detailed proof of (4) and (5) as we can get the result easily

from the definition of the orthogonal matrix, , and (1).

6.3 Kernel and Range

Reference video: https://youtu.be/vMulGT9RiI0 http://youtu.be/9YciT9Bb2B0

We will show that the subset of a domain which maps to zero vector by a linear transformation,

becomes a subspace. We will also show the set of images under any linear transformation forms a subspace in the co-domain.

Finally, we introduce the concept of isomorphism.

 Definition[Kernel] Let  be a linear transformation. The set of all vectors in , whose image becomes  by , is called kernel of  and is denoted by .      That is,      .

Find the  for a linear transformation where .

.

Find the  for a linear transformation , where .

For any ,

and hence, .

 Definition  [one-to-one] For a transformation , if ,               then it is called one-to-one (injective).

 Definition  [onto] For a transformation , if there exist  for any given ,   such that ,  then it is called onto (surjective).

 Theorem  6.3.1 Let  and  are vector spaces and  is a linear transformation.      Then  is one-to-one       if and only if    .

As and  is one-to-one,

is one-to-one.

Let us define a linear transformation  as . Is  one-to-one?

As ,

the only element in this set is .

Hence , and  is one-to-one.

Is a linear transformation  one-to-one if it is defined as

?

Since      ,

the system of linear equations has infinitely many solutions.

Hence,   and by theorem 6.3.1,  is not one-to-one.

① Verify whether a linear transformation is one-to-one or not

False

② Find the Kernel of the linear transformation

Vector space of degree 3 and dimension 1 over Rational Field

Basis matrix:

[   1 –1/3  1/3]             # kernel = span( [1, -1/3, 1/3] ).

Let  be an  matrix. If we define a linear transformation  as ,

then  is a solution space of the system of linear equations .

 Theorem6.3.2 Let  are vector spaces and  is a linear transformation.  Then  is a subspace of .  Hence the subspace  is called a kernel.

is a subspace of .

Find the kernel of  when  with .

Free module of degree 2 and rank 0 over Integer Ring

Echelon basis matrix:

[]                   # kernel has only a Zero vector

 Definition[Isomorphism] For a linear transformation , the set of all  for , is called range of  and is denoted by .  That is,   .   Especially, if  then  is called surjective or onto.   If a linear transformation  is one-to-one and onto, then  and  is called anisomorphism from  to .

Find the range of the linear transformation .

. Note that,  is not surjective.

is not an isomorphism as it is not surjective.

Let  and

. It is easy to see that both  and  are subspaces of .

If we define  as the following linear transformation,

then  is both one-to-one and onto, and hence it is an isomorphism.

 Theorem  6.3.3 For a linear transformation ,   is a subspace of  .

such that ()

is a subspace of .

Let  be an  matrix, if we define a linear transformation  as ,

then  is a column space of .

Let , that is,  be an  matrix 's th column vector.

Then for any vector ,

.

That is, any image can be expressed as a linear combination of column vectors of .

 Theorem  6.3.4 For a linear transformation  defined by a matrix    satisfies the following two properties.   (1)  is one-to-one.  column vectors  of  are linearly independent.  (2)  is onto.  row vectors of  are linearly independent.

(1)  is one-to-one

There is a unique  which satisfies .

column vectors of  are linearly independent.

(2)  is onto

For 's column vectors ,

In RREF, the number of leading ones is .

row rank of  is .

row vectors of  are linearly independent.

Verify the following by using the Sage.

(1) Let  is one-to-one but not onto.

① define a linear transformation

Vector space morphism represented by the matrix:

[1 0 0]

[0 1 0]

Domain: Vector space of dimension 2 over Rational Field

Codomain: Vector space of dimension 3 over Rational Field

② check the surjectivity (onto)

Vector space of degree 3 and dimension 2 over Rational Field

Basis matrix:

[1 0 0]

[0 1 0]

False

③ check the injectivity (one-to-one)

Vector space of degree 2 and dimension 0 over Rational Field

Basis matrix:

[]

True

(2) Let  is onto but not one-to-one.

① define a linear transformation

Vector space morphism represented by the matrix:

[1 0]

[0 1]

[0 0]

Domain: Vector space of dimension 3 over Rational Field

Codomain: Vector space of dimension 2 over Rational Field

② check the surjectivity (onto)

Vector space of degree 2 and dimension 2 over Rational Field

Basis matrix:

[1 0]

[0 1]

True

③ check the injectivity (one-to-one)

 Theorem 6.3.5 Let be an matrix. If is a linear transformation,      is one-to-one    if and only if     is onto.

is one-to-one

There is a unique which satisfies .

In 's RREF, number of leading ones is .

For 's column vectors ,

is onto

Invertible Matrix theorem

 Theorem 6.3.6 [Invertible Matrix Theorem] Let be an matrix, the following statements are all equivalent.    (1) column vectors of are linearly independent.  (2) row vectors of are linearly independent.  (3) has only trivial solution .  (4) For any vector , has a unique solution.  (5) and are column equivalent.  (6) is invertible.  (7)  (8) is not an eigenvalue of .  (9) is one-to-one.  (10) is onto.

6.4 Composition of Linear Transformations and Invertibility

Reference video: https://youtu.be/AfjRc-IxZQk

In this section, we study the composition of  two or more linear transformations as continuous product of matrices.

We also study the geometric properties of linear transformation by connecting inverse functions and inverse matrices.

 Theorem 6.4.1 [Composition of Functions] If both and are linear transformations, then the composition                 is also a linear transformation.

 Theorem 6.4.2 For linear transformations and ,  (1)   is one-to-one implies is one-to-one.  (2)   is onto implies is onto.

(1) If , for all , then .

( is one-to-one)

∴ is one-to-one.

(2) If is onto, then for , there exist such that .

That is, there exist which satisfy .

Since ,  there exist such that

∴ is onto.

For the case of composition of two linear transformations,

the corresponding standard matrix is the product of two standard matrices from each linear transformation.

That is, let , and

has a standard matrix , has standard matrix .

Then the linear transformation has the standard matrix

.

Let the standard matrix of a linear transformation  be .

If an inverse transformation exist, then the standard matrix of is the inverse of the matrix .

Let are linear transformations which rotate and (counterclockwise)

respectively around the origin. The corresponding standard matrices are as follows.

,

As the composition of these two transformations rotates around the origin,

the standard matrix of   is as follows.

.

Also the product of standard matrices of and are as follows.

.

As shown in the picture, find a matrix transformation which transform a circle with radius 1 to the given ellipse.

First we find a transformation which expands 3 times around the -axis, and expands 2 times around the -axis.

Then take a transformation which rotates around the origin.

The first transformation is , and

hence the standard matrices for and the rotation transformation are

.

Therefore, the standard matrix for the composition is the product of two standard matrices.

.

 [Remark] Computer simulation [Matrix Transformation] http://www.geogebratube.org/student/m57556

Similarly a composition of three or more linear transformations,

the standard matrix of the composition is the product of each standard matrix in that operation order.

 Theorem 6.4.3 A function is invertible if and only if is one-to-one and onto.

 Theorem 6.4.4 If a linear transformation is invertible, then  is also a linear transformation.

Inverse transformation of composition of transformation:

 [Remark] Computer simulation [ Scaling (expanding and compressing) transformation]

6.5 *Computer Graphics with Sage

Reference video: http://youtu.be/VV5zzeYipZs

Practice site: http://matrix.skku.ac.kr/Lab-Book/Sage-Lab-Manual-2.htm

Computer graphics plays a key role in automotive design, flight simulation, and game industry.

For example, a 3 dimensional object,  such as automobile, its data (coordinates of points) can be described as a matrix.

If we transform the location of these points, we can redraw the transformed object from the points which are newly generated.

If this transformation is linear, we can easily obtain the transformed data by matrix multiplication.

In this section, we review several geometric transformations which are used in computer graphics.

Geometric meaning of Linear Transformation 1

(Linear Transformation of Polygon’s Image)

By using the Sage, draw a triangle with three vertices , , and ,

a triangle expanded twice, a figure by a shear transformation along the -axis with scale 1, and a triangle which is rotated counterclockwise by .

First of all, in order to define the above linear transformations, we input the following linear transformations by using matrix.

(The matrix_transformation function is activated)

Then, we define appropriate standard matrices to fit the problems’ condition.

Draw a triangle which has three vertices  by using ploygon.

Draw a twice expanded triangle from the given triangle.

Draw a shear transformed figure along the -axis with scale 1 from the given triangle.

Draw a figure which is rotated counterclockwise by  from the given triangle.

Show the above four figures in the same frame.

Geometric meaning of Linear Transformation 2

(Linear Transformation of Line’s Image)

Draw the alphabet letter  on the plane. Then draw figures which expands the original figure twice,

sheer transforms along the -axis with scale 1, and rotates counterclockwise by  .

First of all, in order to define the above linear transformations, we input the following linear transformations by using matrix.

Then, we define appropriate standard matrices to fit the problems’ condition.

Draw an alphabet letter S by using the line function.

Draw a twice expanded letter S from the given figure.

Draw a sheer transformed figure along the -axis with scale 1 from the given S.

Draw a figure which is rotated counterclockwise by  from the given letter S.

Show the above four figures in the same frame.

2015.10.1. Solved by 강민혁, 2015.10.1. Revised by 김원경, 이승열, Final OK by SGLee

Chapter6 p.g224

Verify that , where , is a linear transformation and find for .

Sol)

Let .

1)  (∵)

⇒

2)

⇒

Answer)  is a linear transformation from .               ■

for

Answer)            ■

Double check by Sage)

Solution)

■

Note : 표준기저를 이용하여 문제를 구하는 것이므로 의도에 맞게 풀어야 합니다.

Let a linear transformation  satisfy the following conditions:

.

(1) Evaluate .

(2) Evaluate .

Solution)

1)

2)

■

Note : Linear Transform의 성질만 이용하여 문제를 간단히 풀 수 있습니다.

Sol)

Ans:                    ■

Double checked by Sage)

결과:

[3]

[1]

[1/2*sqrt(3) - 3/2]

[3/2*sqrt(3) + 1/2]                              ■

Check whether the given matrix is an orthogonal matrix. If that is the case, find the inverse matrix.

Solution)

===

Similarly

∴ = : ( is an orthogonal matrix), =

■

Double check by Sage)

```AT*A = [1 0]
[0 1]        ■  ```

```Problem 6

For each given linear transformation, find the kernel and range. Also determine whether it is bijective or not.

(1)
(2)

Sol)
(1) 선형변환 의 표준행렬 :

⇒ 는 가역행렬이다  의 행벡터, 열벡터 모두 일차독립이다  Im.
=0 이다  는 단사이다. Im이다 는 전사이다.

, Im , bijective ■

(2) 선형변환 의 표준행렬 :

는 비가역 행렬이다.
선형변환 를 하면 평면위의 모든 점들이 () 으로 이동한다.  Im ().
=0 이다  는 단사이다. Im()이다  는 전사가 아니다.

, Im , injective ■Chapter6 p.g225
Let  and  are defined as follows:

.
(1) Find the standard matrix for each  and .
(2) Find the standard matrix for each  and .
Sol)
(1)

■
(2)

■

Double check by Sage) ```

[T1]=

[ 4  0  0]

[-2  1  0]

[-1 -3  0]

[T2]=

[ 1  2  0]

[ 0  0 -1]

[ 4  0 -1]

[T2*T1]=

[ 4  8  0]

[-2 -4 -1]

[-1 -2  3]

[T1*T2]=

[ 0  2  0]

[ 1  3  0]

[17  3  0]

Comment : Ch.6의 2번문제와 마찬가지로 선형변환 ,는 3차원이므로 가장 간단한 단위직교벡터를 이용하여 표준행렬을 찾았고, Sage의 계산결과와 같음을 확인 할 수 있었다.

Chapter6 p225

Sol)

■

Double check by Sage)

(S dot T)(x)=

(x_1, x_2) |--> (x_1 + 2*x_2, -x_1 - x_2)

Answer the following questions.
(1) Find the dimension of the null space of the following matrix by using the Sage.

(2) Let  be a linear transformation corresponding to the above matrix . Determine whether  is in the range of  by using the Sage.

Sol.

(1)

■

(2)

is a linear combination of Columns of A.

is in the range of .                                       ■

Note. ​로 이뤄진 선형변환의 치역에 벡터 ​가 있으려면, 벡터 ​​가 행렬 ​​의 열벡터들의 일차결합으로 이루어져야 한다.

Let . Find  by using the Sage.

Sol)

[cos(t)  -sin(t)  -x0*cos(t)+y0*sin(t) + x0]

[sin(t)  cos(t)  -x0*sin(t) -y0*cos(t) + y0]

[0              0                      1]

Ans)   [x*cos(t) - x0*cos(t) - y*sin(t) + y0*sin(t) + x0]

[x*sin(t) - x0*sin(t) + y*cos(t) - y0*cos(t) + y0]

[                                             1]

Note) (x0, y0, 1)을 기준으로  x축과 y축 방향으로 각각   만큼 이동하였다.

이동한 값을 제곱해서 더하면   이 된다.  (x, y, 1)는 theta 값에 따라서 (x0, y0, 1)을 기준으로 두 점 사이의 거리를 반지름 으로 하는 원 위의 점이 된다.

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