Basics of Linear Algebra

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Basics of Linear Algebra

• A review?

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Matrix

• Mathematical term essentially corresponding to an
  array
• An arrangement of numbers into rows and columns.
  
• Each row is the same length
• Each column is the same length
• Usually, we specify a position in the matrix as
  row, column

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Vector

• A one dimensional matrix
• One row (or one column)
• We can treat it as a special case of a matrix

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Matrix operations

• Matrix (vector) addition/subtraction
• Add/subtract the corresponding elements in two
  matrices (vectors) of exactly the same size and
  shape.
• If A and B
• What is A B?
• Matrices can only be added or subtracted if they
  are exactly the same size and shape.

• 17 45
• 44 47
• 62 25 17

• 120 115
• 230 204
• 301 300 320

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Multiplication

• If A is an m x n matrix and B is an n x p matrix,
  then the product AB is an m x p matrix whose
  elements are defined by
• cij ?aikbkj
• That is, sum the term by term products of the
  elements in row I of A column j of B.

n
k 1
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Transposition

• If A is m x n, the transpose of A is n x m.
• The rows become columns and the columns become
  rows.
• This is sometimes needed to put things into a
  form that is compatible for multiplication.

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Properties

• The identity matrix has 1s on the main diagonal
  and 0s elsewhere.
• Multiplication by the identity matrix yields the
  original matrix. i.e. AI IA A
• The size of the identity matrix is made to be
  compatible for the operation intended.
• The zero matrix has 0 in every position.
• If A, B, C are of appropriate sizes, then
• A(BC) (AB)C
• A(BC) AB AC
• (AB)C AC BC

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Matrix inverse

• The inverse (A-1) is defined such at A A-1 is I.
• Not every matrix has an inverse. If no inverse
  exists, then the matrix is called singular (non
  invertible)
• If A is nonsingular, so is A-1
• If A, B are nonsingular, then AB is also non
  singular and (AB) -1 B -1A -1 (Note reversed
  order.)
• If A is nonsingular, then so is its transpose and
• (AT ) -1 (A-1)T

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Vectors and Vector Spaces

• A vector with 2 elements (a 2-vector) is written
  as( ).
• The vector is represented by a line in a plane,
  starting at the origin and ending at the point
  x,y. For x2 and y1
• The length of the vector is calculated using the
  Pythagorean theorem vector v x2 y2

x y
(2,1)
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Vector operations

• Addition and subtraction consist of adding or
  subtracting the corresponding elements. Only
  vectors of the same size can be added/subtracted.
• What does the sum of two vectors look like in the
  coordinate system?

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Vector angles

• There is an angle between any two vectors in the
  coordinate system. One way of comparing the
  vectors is to measure the angle between them.
• Cos ?
• Where X is the length, or magnitude, of X
• X ( ) Y ( )

x1x2 y1y2 X Y
x1 y1
x2 y2
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Dot product and cosine

• The dot product of vectors X, Y is defined as
• X Y x1x2 y1y2
• So, Cos ?
• If the cosine ? is 0, the vectors are at right
  angles.

X Y X Y
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Larger dimensions

• It is easy to visualize vectors in two space, but
  larger dimensions are also useful. We cannot
  draw them so easily, but the properties of
  length, distance, etc. remain interesting.
• We can draw 3-vectors.
• Note that we cannot express the x axis in terms
  of the y axis. In 3-space, we cannot express any
  of x, y, z in terms of just the others. The
  three axes define the space.
• When we look at the relationships between vectors
  that define index terms or queries, the
  relationship between them tells us whether they
  represent totally unrelated information, or
  information that is more or less related.

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• Reference for this review
• Introductory Linear Algebra with Applications
• Bernard Kolman
• Macmillan Publishing, 1984
• Chapters 1 and 3.