??????12 : Dot Product

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??????12 Dot Product Matrix Operation

• ??????? (Kuang-Chi Chen)
• chichen6_at_mail.tcu.edu.tw

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Linear Equations and MatricesDot Product and
Matrix Multiplication
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Dot Product and Matrix Multiplication

• 1.3 Dot Product and Matrix Multiplication
• Dot Product
• The dot product or inner product of the
  n-vectors a and b

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Example 1

• E.g. 1 the dot product of u and v
• uv (1)(2) (-2(3) (3)(-2) (4)(1) -6

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Example 2

• Example 2
• Let and . If
  ab -4 ,
• find x
• ab 4x 2 6 -4
• 4x 8 -4
• x -3

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Example 3

• E.g. 3 Computing a course average
• wg (.10)(78) (.30)(84) (.30)(62)
  (.30)(85)
• 77.1

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Matrix Multiplication

• Matrix Multiplication
• The product of A and B, denoted by AB C,
• where
• thus

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(contd)

• That is,
• 
  colj(B)
• A
  B AB
• m?p
  p?n m?n

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Example 4

• Example 4

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Example 5

• E.g. 5 Compute the (3, 2) entry of AB

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Example 6 Linear System in Matrix Form

• E.g. 6 Linear system in matrix form
• 
  - a linear system
• 
  - a matrix form

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Example 7 Find x and y

• E.g. 7 Find x and y

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AB and BA

• AB and BA
• - BA may not be defined, while BA is defined
• - AB and BA are of different sizes (A2?3 , B3?2)
• - AB and BA are both of the same size and they
  may be equal
• - AB and BA are both of the same size and they
  may not be equal (see example)

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Example 10 AB ? BA

• E.g. 10 - AB ? BA

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Example 11 the 2nd column of AB

• E.g. 11 - the second column of AB

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Remark

• Remark inner product and matrix multiplication
• If u and v are n-vectors
• - Both are n?1, then uv uTv
• - Both are 1?n, then uv uvT
• - u is 1?n and v is n?1, then uv uv .

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Example 12 Inner Product Matrix Multiplication

• E.g. 12 inner product and matrix multiplication

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Matrix-vector Product

• The matrix-vector product written in terms of
  columns

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(contd)
- A linear combination of the columns of A
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Example 13

• Example 13

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Example 14

• E.g. 14

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Linear Systems

• Linear Systems
• Define

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Linear Systems (contd)

• We have
• i.e., Ax b
• Here, A is the coefficient matrix (design matrix).

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Augmented Matrix

• Augmented Matrix

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Example 15 Augmented Matrix

• E.g. 15

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Example 16 Augmented Matrix

• Example 16

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Linear Equations and MatricesProperties of
Matrix Operations
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Properties of Matrix Operations

• 1.4 Properties of Matrix Operations
• Theorem 1.1 Properties of matrix addition
• A B B A
  commutative law
• A (B C) (A B) C associative
  law
• A O A , (O zero matrix) identity
  law
• A D O , A (-A) O additive inverse

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Example 1 Zero Matrix

• E.g. 1
• - The 2?2 zero matrix
• - The 2?3 zero matrix

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Example 2

• Example 2
• A (-A) O

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Example 3 Subtraction

• E.g. 3 Subtraction

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Properties of Matrix Multiplication

• Theorem 1.2 Properties of matrix multiplication
• (AB)C A(BC) associative
  law
• A(B C) AB AC distributive
  law
• (A B)C AC BC .
• No commutative law !

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Example 4 (AB)C A(BC)

• The of multiplications to
  compute
• A(BC) is 3?4?3 2?3?3 36 18
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• The of multiplications to
  compute
• (AB)C is 2?3?4 2?4?3 24 24
  48.

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Example 5

• E.g. 5 A(B C) AB AC

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Identity Matrix of Order n

• Definition Identity matrix of order n
• thus, In A A or A In A .

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Example 6

• E.g. 6 Identity matrix
• I2A A AI3 A

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Powers of A Matrix

• Powers of an n?n matrix
• Definition
• thus, A0 In ,
• ApAq Apq ,
• (Ap)q Apq ,
• (AB)p ? ApBp . (why? No commu. law)

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Example 7 No Cancellation Law

• E.g. 7 The cancellation law doesnt hold.
• We know ab ac and a ? 0 ? b c but it
  doesnt hold in matrix. (see example)

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Example 8

• Example 8
• ,
  but B? C .

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Example 9 Markov Chain

• E.g. 9 Example of Markov chain
• 
  Initial distribution
• 
  of the market

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(contd)

• x2 Ax1 A(Ax0) A2x0 .
• Let x0 a, bT ,
• determine a and b so that the market is stable
• i.e., Ax0 x0 .

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(contd)
We have x0 a, bT , a b 1, and Ax0 x0 ,
i.e., and a
1 b ,
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Properties of Scalar Multiplication

• Theorem 1.3 - Properties of scalar multiplication
• r(sA) (rs)A
  (like-associative law)
• r(A B) rA rB
  (like-distributive law)
• (r s)A rA sA
• A(rB) r(AB) (rA)B .

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Example 10

• Example 10
• r -2 ,

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Properties of Transpose

• Theorem 1.4 Properties of transpose
• (AT)T A (double
  transpose law)
• (A B)T AT BT
• (AB)T BTAT (Check? If A3?2 , B2?3)
• (rA)T rAT .

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Example 11 Transpose

• E.g. 11 Transpose

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Symmetry

• Definition Symmetric
• Let A aijn?n ,
• If AT A , (i.e., aij aji ), then A is
  symmetric.

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Example 12 Symmetry

• E.g. 12 Symmetry

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Linear Equations and MatricesMatrix
Transformation
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Matrix Transformation

• 1.5 Matrix Transformation
• R2 denotes the set of all 2-vectors
• R3 denotes the set of all 3-vectors
• Represented geometrically as directed line
  segments in a rectangular coordinate system

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Vector Space 2- 3-vectors

• Let

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Example 1 n-Vectors

• Example 1

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Definition of Mapping Function

• If A is an m?n matrix and u is an n-vector, then
  Au is an m-vector
• - Mapping Rn into Rm
• A mapping function f Rn ? Rm defined by f(u)
  Au
• - f(u) the image of u
• - the range of f the set of all images of the
  vectors in Rm
• - f the mapping function

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Example 2 Image

• Example 2

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More Examples of Image
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Example 3

• Example 3

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Solution of Example 3
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Example 4 Reflection

• E.g. 4 Reflection
• Let f R2 ? R2 be the matrix transformation
  defined by
• The reflection w.r.t. the axis in R2 .

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Example 5 Projection

• Example 5 Projection (3-dim onto 2-dim)
• Let f R3 ? R2 be the matrix transformation
  defined by
• then

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contd

• More Projection (3-dim onto xy-plane)
• Let f R3 ? R3 be the matrix transformation
  defined by
• then

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Example 6

• Example 6
• Let f R3 ? R3 be the matrix transformation
  defined by
• where
  r is a real number,
• so we have f(u) ru .
• If r gt 1, then dilation stretches a vector
• if 0 lt r lt 1, then contraction shrinks a
  vector.

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Example 7

• Example 7
• - Rotate every point in R2 counterclockwise
  through an angle ? about the origin of a
  rectangular coordinate system

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(contd)

• Let x r cos ? ,
• y r sin ? ,
• x r cos (? ?) ,
• y r sin (? ?) ,
• since x r cos ? cos ? r sin ? sin ?
  ,
• and y r sin ? cos ? r cos ? sin ? .

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(contd)

• so x x cos ? y sin ? ,
• and y x sin ? y cos ? .
• Thus,
• where u
  (x, y)T.

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(contd)

• Similarly, x x cos ? y sin ? ,
• and y -x sin ? y cos ? .
• Thus,
• where u
  (x, y)T.
• Moreover, f is the inverse function of f .

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(contd)

• Note