Chapter 2�Matrices and Linear Transformations

大葉大學 資訊工程系

黃鈴玲

Linear Algebra

2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

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• a_ij: the element of matrix A in row i and column j.

​

Definition

Two matrices are equal if they are of the same size and if their corresponding elements are equal.

Thus A = B if a_ij = b_ij ∀ i, j.

(∀ 代表 for every, for all)

Addition of Matrices

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Definition

Let A and B be matrices of the same size.

Their sum A + B is the matrix obtained by adding together the corresponding elements of A and B.

The matrix A + B will be of the same size as A and B.

If A and B are not of the same size, they cannot be added, and we say that the sum does not exist.

Example

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Solution (自行練習)

(2) A and C are not of the same size, A + C does not exist.

Scalar Multiplication of matrices

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Definition

Let A be a matrix and c be a scalar. The scalar multiple of A by c, denoted cA, is the matrix obtained by multiplying every element of A by c. The matrix cA will be the same size as A.

Negation and Subtraction

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Definition

The matrix (−1)C is written –C and is called the negative of C.

We now define subtraction in terms of addition and scalar multiplication. Let

A – B = A + (–1)B

Matrix Multiplication

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Examples

Example 1

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無法相乘�∴AB does not exist.

Sol.

Note. In general, AB≠BA. 如此題之BA存在

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Definition

Let the number of columns in a matrix A be the same as the number of rows in a matrix B. The product AB then exists.

If the number of columns in A does not equal the number of row B, �we say that the product does not exist.

Let A: m×n matrix, B: n×k matrix,

The product matrix C=AB has elements

C is a m×k matrix.

Example 2

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Sol.

Note. In general, AB≠BA.

BA does not exist.

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隨堂作業：5(f)

Size of a Product Matrix

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If A is an m × r matrix and B is an r × n matrix, then AB will be an m × n matrix.

B

r × n

= AB

m × n

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Definition

A zero matrix is a matrix in which all the elements are zeros.

A diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros.

An identity matrix is a diagonal matrix in which every diagonal element is 1.

Special Matrices

Theorem 2.1

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Let A be m × n matrix and O_mn be the zero m × n matrix. Let B be an n × n square matrix. O_n and I_n be the zero and identity n × n matrices. Then

A + O_mn = O_mn + A = A

BO_n = O_nB = O_n

BI_n = I_nB = B

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(a) A: m×n, B: n×r� Let the columns of B be the matrices B₁, B₂, …, B_r. � Write B=[B₁ B₂ … B_r]. � Thus AB=A[B₁ B₂ … B_r]=[AB₁ AB₂ … AB_r].

Matrix multiplication in terms of columns

Example

⇒

and

⇒

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(b) � A: m×n, B: n×1, where A=[A₁ A₂ … A_n] and .� �� We get .

Matrix multiplication in terms of columns

Example

⇒

⇒

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Partitioning of Matrices

A matrix can be subdivided into a number of submatrices.

Example

where

⇒

Example

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Example 5

Let

⇒

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Under this partition A is interpreted as a 2×2 matrix. For the product AB�to be exist, B must be partitioned into a matrix having two rows.

Homework

• Exercise 2.1:�5, 8, 11, 17, 21

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Solution

2.2 Algebraic Properties of Matrix Operations

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Theorem 2.2 -1

Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.

Properties of Matrix Addition and scalar Multiplication

1. A + B = B + A Commutative property of addition

2. A + (B + C) = (A + B) + C Associative property of addition

3. A + O = O + A = A (where O is the appropriate zero matrix)

4. c(A + B) = cA + cB Distributive property of addition

5. (a + b)C = aC + bC Distributive property of addition

6. (ab)C = a(bC)

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Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.

Properties of Matrix Multiplication

1. A(BC) = (AB)C Associative property of multiplication

2. A(B + C) = AB + AC Distributive property of multiplication

3. (A + B)C = AC + BC Distributive property of multiplication

4. AI_n = I_nA = A (where I_n is the appropriate zero matrix)

5. c(AB) = (cA)B = A(cB)

Note: AB≠ BA in general. Multiplication of matrices is not commutative.

Theorem 2.2 -2

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Proof of Thm 2.2 (A+B=B+A)

Consider the (i,j)th elements of matrices A+B and B+A:

∴ A+B=B+A

Example 2

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Sol.

A B C = D�2×2 2×3 3×1 2×1

ABC = (AB)C = A(BC)

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In algebra we know that the following cancellation laws apply.

• If ab = ac and a ≠ 0 then b = c.
• If pq = 0 then p = 0 or q = 0.

However the corresponding results are not true for matrices.

• AB = AC does not imply that B = C.
• PQ = O does not imply that P = O or Q = O.

Caution

Example

Powers of Matrices

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Theorem 2.3

If A is an n × n square matrix and r and s are nonnegative integers, then

1. A^rA^s = A^r+s.

2. (A^r)^s = A^rs.

3. A⁰ = I_n (by definition)

Definition

If A is a square matrix, then

Example 3

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Solution

Systems of Linear Equations

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A system of m linear equations in n variables as follows

Let

We can write the system of equations in the matrix form

​

AX = B

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Solutions to Systems of Linear Equations

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Consider a homogeneous system of linear equations AX=0. �Let X₁ and X₂ be solutions. Then� AX₁=0 and AX₂=0

⇒ A(X₁ + X₂) = AX₁ + AX₂ = 0 ⇒ X₁ + X₂ is also a solution

Note. �The set of solutions to a homogeneous system of linear equations�is closed under addition and under scalar multiplication. It is a subspace.

If c is a scalar,�⇒ A(cX₁) = cAX₁ = 0 ⇒ cX₁ is also a solution

Example 5

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Consider the following homogeneous system of linear equations.

It can be shown that there are many solutions,� x₁=2r, x₂=3r, x₃ = r.

The solutions are vectors in R³ of the form (2r, 3r, r) or r(2, 3, 1). The solutions form a subspace of R³ of dimension 1.

Note. x₁=0, …, x_n = 0, is a solution to every homogeneous system.�⇒ The set of solutions to every homogeneous system passes through the origin.

隨堂作業：36(a)

Solutions to Nonhomogeneous systems

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Let AX=Y be a nonhomogeneous system of linear equations, so Y≠0. �Let X₁ and X₂ be two solutions. Then� AX₁=Y and AX₂=Y

⇒ A(X₁ + X₂) = AX₁ + AX₂ = 2Y ⇒ X₁ + X₂ is not a solution.

(可省略) If c is a scalar,�⇒ A(cX₁) = cAX₁ = cY ⇒ cX₁ is not a solution.

Exercise 41

Show that a set of solutions to a system of nonhomogeneous linear equations is not closed under addition or under scalar multiplication, and that it is thus not a subspace of Rⁿ.

Proof

⇒The set of solutions is not a subspace of Rⁿ.

Example

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Consider the following nonhomogeneous system of linear equations.

The general solution of this system is (2r+4, 3r+2, r).

(2r+4, 3r+2, r) = r(2, 3, 1)+(4, 2, 0)

Note that r(2, 3, 1) is the general solution of the corresponding homogeneous system. The vector (4, 2, 0) is the specific solution to the nonhomogeneous system corresponding to r=0.

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Idempotent and Nilpotent Matrices�(Exercise 24~30)

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Definition

1. A square matrix A is said to be idempotent (等冪) if A²=A.
2. A square matrix A is said to nilpotent (冪零) if there is a � positive integer p such that A^p=0. The least integer p such that �A^p=0 is called the degree of nilpotency of the matrix.

Example

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Homework

• Exercise 2.2:�6, 13, 21, 25, 27, 28, 29, 32, 36, 40, 41

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Homework

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2.3 Symmetric Matrices

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Definition

The transpose (轉置) of a matrix A, denoted A^t, is the matrix whose columns are the rows of the given matrix A.

Theorem 2.4 Properties of Transpose

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Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.

1. (A + B)^t = A^t + B^t Transpose of a sum

2. (cA)^t = cA^t Transpose of a scalar multiple

3. (AB)^t = B^tA^t Transpose of a product

4. (A^t)^t = A

Theorem 2.4 Properties of Transpose

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Proof for 3. (AB)^t = B^tA^t

Symmetric Matrix

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Definition

A symmetric matrix is a matrix that is equal to its transpose.

Example

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Example 3

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Let C = aA+bB, where a and b are scalars.

C^t = (aA+bB)^t

= (aA)^t + (bB)^t by Thm 2.4 (1)

= aA^t + bB^t by Thm 2.4 (2)

= aA + bB since A and B are symmetric

= C

Thus C is symmetric.

Proof

Let A and B be symmetric matrices of the same size. Let C be a linear combination of A and B. Prove that the product C is symmetric.

If and only if

• Let p and q be statements.�Suppose that p implies q (if p then q), written p ⇒ q,�and that also q ⇒ p, we say that��“p if and only if q” � (若且唯若)� (通常簡寫為 iff )� (也說成: p is necessary and sufficient for q )

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Example 4

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*We have to show (a) AB is symmetric ⇒ AB = BA, � and the converse, (b) AB is symmetric ⇐ AB = BA.

(⇒) Let AB be symmetric, then

AB= (AB)^t by definition of symmetric matrix

= B^tA^t by Thm 2.4 (3)

= BA since A and B are symmetric

(⇐) Let AB = BA, then

(AB)^t = (BA)^t

= A^tB^t by Thm 2.4 (3)

= AB since A and B are symmetric

Proof

Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA.

Example 3 相關題目

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Proof

Let A be a symmetric matrix. Prove that A² is symmetric.

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Definition

Let A be a square matrix. The trace of A, denoted tr(A) is the sum of the diagonal elements of A. Thus if A is an n × n matrix.

tr(A) = a₁₁ + a₂₂ + … + a_nn

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Trace of a matrix

Theorem 2.5 Properties of Trace

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Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.

1. tr(A + B) = tr(A) + tr(B)

2. tr(AB) = tr(BA)

3. tr(cA) = c⋅ tr (A)

4. tr(A^t) = tr(A)

Since the diagonal element of A + B are (a₁₁+b₁₁), (a₂₂+b₂₂), …, (a_nn+b_nn), we get

tr(A + B) = (a₁₁ + b₁₁) + (a₂₂ + b₂₂) + …+ (a_nn + b_nn)

= (a₁₁ + a₂₂ + … + a_nn) + (b₁₁ + b₂₂ + … + b_nn)

= tr(A) + tr(B).

Proof of (1)

Example of (2) tr(AB)=tr(BA)

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Matrices with Complex Elements

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The element of a matrix may be complex numbers. A complex number is of the form

z = a + bi

Where a and b are real numbers and �a is called the real part and b the imaginary part (虛部) of z.

Example 7

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Solution

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Definition

(i) The conjugate of a matrix A is denoted A and is obtained by � taking the conjugate of each element of the matrix.

(ii) The conjugate transpose of A is written and defined by A^*=A ^t.

(iii) A square matrix C is said to be hermitian if C=C^*.

Example (i), (ii)

Example (iii)

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Homework

• Exercise 2.3:�2, 5, 11, 15, 23

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2.4 The Inverse of a Matrix

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Definition

Let A be an n × n matrix. If a matrix B can be found such that AB = BA = I_n, then A is said to be invertible (可逆) and B is called the inverse (反矩陣) of A. If such a matrix B does not exist, then A has no inverse. (denote B = A⁻¹, and A^−k=(A⁻¹)^k )

Thus AB = BA = I₂, proving that the matrix A has inverse B.

Theorem 2.7

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If a matrix has an inverse, that inverse is unique.

Proof

Let B and C be inverses of A.

Thus AB = BA = I_n, and AC = CA = I_n.

Multiply both sides of the equation AB = I_n by C.

C(AB) = CI_n

(CA)B = C

I_nB = C

B = C

Thus an invertible matrix has only one inverse.

Let A be an invertible n×n matrix�How to find A^-1?

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We shall find A⁻¹ by finding X₁, X₂, …, X_n.

Solve these systems by using Gauss-Jordan elimination:

Note. 若是最後沒有變成 [I_n:B] 的形式，則 A⁻¹不存在。

Gauss-Jordan Elimination for finding the Inverse of a Matrix

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Let A be an n × n matrix.

1. Adjoin the identity n × n matrix I_n to A to form the matrix [A : I_n].

2. Compute the reduced echelon form of [A : I_n].

If the reduced echelon form is of the type [I_n : B], then B is the inverse of A.

If the reduced echelon form is not of the type [I_n : B], in that the first n × n submatrix is not I_n, then A has no inverse.

An n × n matrix A is invertible if and only if its reduced echelon form is I_n.

Example 2

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Solution

隨堂作業：4(a), 14

Example 3

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Determine the inverse of the following matrix, if it exist.

Solution

There is no need to proceed further.

The reduced echelon form cannot have a one in the (3, 3) location.

The reduced echelon form cannot be of the form [I_n : B].

Thus A^–1 does not exist.

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Properties of Matrix Inverse

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Let A and B be invertible matrices and c a nonzero scalar, Then

​

​

​

​

Proof

1. By definition, AA⁻¹=A⁻¹A=I.

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Solution

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Theorem 2.8

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Let AX = Y be a system of n linear equations in n variables. �If A^–1 exists, the solution is unique and is given by X = A^–1Y.

Proof

(X = A^–1Y is a solution.)�Substitute X = A^–1Y into the matrix equation.

AX = A(A^–1Y) = (AA^–1)Y = I_nY = Y.

�(The solution is unique.)

Let X₁ be any solution, thus AX₁ = Y. Multiplying both sides of this equation by A^–1 gives

A^–1AX₁= A^–1Y �I_nX₁ = A^–1Y�X₁ = A^–1Y.

Example 5

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Solution

Elementary Matrices

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Definition

An elementary matrix is one that can be obtained from the identity matrix I_n through a single elementary row operation.

Example

Elementary Matrices

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一個矩陣做 elementary row operation，�相當於在左邊乘一個對應的 elementary matrix。

Notes for elementary matrices

• Each elementary matrix is square and invertible.

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Example

• If A and B are row equivalent matrices and A is invertible, then B is invertible.

Proof

If A ≈ … ≈ B, then

B=E_n … E₂ E₁ A for some elementary matrices E_n, … , E₂ and E₁.

So B⁻¹ = (E_n … E₂ E₁A)⁻¹ =A⁻¹E₁⁻¹ E₂⁻¹ … E_n⁻¹.

Homework

• Exercise 2.4:�4, 7, 14, 15, 19, 21, 28

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(後面的 section 2.5~2.7 都跳過)

（求2×2之反矩陣，此公式較快）

Homework

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2.5 Matrix Transformations, Rotations, and Dilations

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A function, or transformation, is a rule that assigns to each element of a set a unique element of another set.

We will be especially interested in linear transformations, which are transformations that preserve the mathematical structure of a vector space.

Consider the function f(x) =3x²+4.

domain (定義域) of the function: the set of all possible x

f(2)=16, we say that the image of 2 is 16.

Extend these ideas to functions between vector spaces

We usually use the term transformation rather than function in linear algebra.

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Consider the transformation T maps R³ into R² defined by

T(x, y, z) = (2x, y−z)

The domain of T is R³ and we say the codomain (對應域) is R².

T(1, 4, −2) = (2, 6) ⇒ The image of (1, 4, −2) is (2, 6).

Convenient representations:

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Definition

A transformation T of Rⁿ into R^m is a rule that assigns to each vector u in Rⁿ a unique vector v in R^m. Rⁿ is called the domain of T and R^m is the codomain. We write T(u) = v; v is the image of u under T. The term mapping is also used for a transformation.

Dilation(擴張) and Contraction(收縮)

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Consider the transformation , where r > 0.

Figure 2.11

This equation can be written as the following useful matrix form.

Reflection(反射)

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Consider the transformation . T is called a reflection.

This equation can be written as the following useful matrix form.

Figure 2.12

Matrix transformations

Every matrix defines a transformation. Let A be a matrix and x be a column vector such that Ax exists. Then A defines the matrix transformation T(x) = Ax.

For example, defines ��

⇒

and

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Figure 2.14 Matrix transformations.

, written as

, written as

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Definition

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Let A be an m×n matrix. Let x be an element of Rⁿ written in a column matrix form. A defines a matrix transformation T(x)=Ax of Rⁿ into R^m. The vector Ax is the image of x. The domain of the transformation is Rⁿ and codomain is R^m.

We write T: Rⁿ → R^m if T maps Rⁿ into R^m.

Geometrical Properties:�Matrix transformations maps line segments (線段) into line segments (or points). If the matrix is invertible (可逆) the transformation also maps parallel lines into parallel lines.

Example 1

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Consider the transformation T: R² → R² defined by the matrix� � . Determine the image of the unit square under this ��transformation.

Sol.

The unit square is the square whose vertices are the points

Since

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The images are:

Figure 2.15

The square PQRO is mapped into the parallelogram P’Q’R’O.

Composition(合成) of Transformations

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Figure 2.16

Consider the matrix transformations T₁(x)=A₁x and T₂(x)=A₂x. �The composite transformation T=T₂ ° T₁ is given by� T(x) = T₂(T₁(x)) = T₂ (A₁x) =A₂A₁x

Thus T is defined by the matrix product A₂A₁.� T(x) = A₂A₁x

隨堂作業：10(a)

Homework

• Exercise 2.5:�1, 10

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2.6 Linear Transformations

A vector space has two operations: addition and scalar multiplication.

Consider the matrix transformation T(u)=Au.

T(u+v) = A(u+v) = Au+Av = T(u)+T(v)

Definition �Let u and v be vectors in Rⁿ and let c be a scalar. A transformation �T: Rⁿ → R^m is said to be a linear transformation if �T(u + v) = T(u) + T(v) and T(cu) = cT(u).

T(cu) = A(cu) = cAu = cT(u)

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Example 1

Prove that the following transformation T: R² → R² is linear.

T(x, y) = (x − y, 3x)

“×”: Let c be a scalar.

T(c(x₁, y₁)) = T(cx₁, cy₁) = (cx₁ − cy₁, 3cx₁)

= c(x₁ − y₁, 3x₁) � = cT(x₁, y₁)

Thus T is linear.

Note A linear transformation T: U → U is called a linear operator.

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Example 2

Show that the following transformation T: R³ → R² is not linear.

T(x, y, z) = (xy, z)

T is not linear.

and T(x₁, y₁, z₁) + T(x₂, y₂, z₂) = (x₁y₁, z₁) + (x₂y₂, z₂)

Thus, in general� T((x₁, y₁, z₁) + (x₂, y₂, z₂)) ≠ T(x₁, y₁, z₁) + T(x₂, y₂, z₂).

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隨堂作業：4(b)

Example 3

Determine a matrix A that describes the linear transformation

T can be written as

Why? See the next page.

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Matrix Representation

Let T: Rⁿ → R^m be a linear transformation. {e₁, e₂, …, e_n} be the standard basis of Rⁿ, and u be an arbitrary vector in Rⁿ.

Express u in terms of the basis, u = a₁e₁+a₂e₂+ … + a_ne_n.

Since T is a linear transformation,

T(u) = T(a₁e₁+ a₂e₂ + … + a_ne_n) = T(a₁e₁) + T(a₂e₂) + …+ T(a_ne_n)� = a₁T(e₁) + a₂T(e₂) + …+ a_nT(e_n)�

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Thus the linear transformation T is defined by the matrix

A = [ T(e₁) … T(e_n) ].

A is called the standard matrix of T.

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Example 4

The transformation defines a reflection in the line y = −x.�It can be shown that T is linear. Determine the standard matrix of this transformation. Find the image of .

T can be written as

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Figure 2.22

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Homework

• Exercise 2.6:�1, 4, 12

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(2.7~2.9節跳過)