Linear Algebra

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Linear Algebra

• The Algebra of Linear Equations and Linear
  Transformations

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Describing a Line

• Start and Endpoint
• START at 0,0
• END at 200 east, 160 north
• Start, Direction, Length
• START at 0,0
• Turn 115 from north
• Move forward by 160

END
START
angle
START
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Vector

• A quantity characterized by a magnitude and
  direction
• Can be represented by an arrow, where magnitude
  is the length of the arrow and the direction is
  given by slope of the line

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Conventions

• Vector quantities denoted as v or
• We will use column format vectors
• Each vector is defined with respect to a set of
  basis vectors (which define a co-ordinate system).

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Row vs. Column Formats

• Both formats, though appearing equivalent, are in
  fact fundamentally different
• be wary of different formats used in textbooks

row format
column format
transposed
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Co-ordinate Systems

• By convention we usually employ a Cartesian
  basis
• basis vectors are mutually orthogonal and unit
  length
• basis vectors named x, y and z
• We need to define the relationship between the 3
  vectors there are 2 possibilities
• right handed systems z comes out of page
• left handed systems z goes into page
• This affects direction of rotations and
  specification of normal vectors

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Cartesian co-ordinate System
RHS
LHS
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Vectors Points

• vectors represent directions
• points represent positions
• Both are meaningless without reference to a
  coordinate system
• vectors require a set of basis vectors
• points require an origin and a vector space

both vectors equal
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Vector Addition

• Addition of vectors follows the parallelogram law
  in 2D and the parallelepiped law in higher
  dimensions

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Vector Addition
u
OR
v
u
v
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Vector Multiplication by a Scalar

• Multiplication by a scalar scales the vectors
  length appropriately (but does not affect
  direction)

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Subtraction
-v
u
-v
Can be seen as an addition of u (-1v)
-v
u
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Linear Combinations

• The linear combination of a set of vectors is the
  sum of scalar multiples of those vectors
• Fixing vectors vi yields an infinite number of u
  depending on the scalars ai.
• The set u is called the span of the vectors vi
• The vectors vi are term basis vectors for the
  space.
• If none of the vi can be created as a linear
  combination of the others, the vectors vi are
  said to be linearly independent.
• All linear combinations contain the zero vector.

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Linear Combinations

• Linear combinations of 1 vector an infinite
  line

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Linear Combinations

• Linear combinations of 2 vectors a plane

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Linear Combinations

• The linear combination of 3 vectors a 3D
  volume.
• The 3D Cartesian coordinate system employs the
  well-known 3D co-ordinate basis x, y and z

The vector v here is a linear combination of the
basis vectors x, y and z
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Vector Magnitude

• The magnitude or norm of a vector of dimension
  n is given by the standard Euclidean distance
  metric

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Normal Vectors

• Vectors of length 1 (unit vectors) are often
  termed normal vectors.
• When we only wish to describe direction we use
  normalised vectors.
• For this and other reasons, we often need to
  normalise a vector
• e.g.

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Dot Product

• Dot product (inner product) is defined as
• Note
• Therefore we can redefine magnitude in terms of
  the dot-product operator
• Dot product operator is commutative and
  associative.

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Dot Product

• The Dot Product can also be obtained from the
  following equation
• where q is the angle between the two vectors
• So, if we know the vectors u and v, then the dot
  product is useful for finding the angle between
  two vectors.
• Note that if we had already normalised the
  vectors u and v then it would simply be

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Dot Product Properties

• If one of the vectors is normal, the dot product
  defines the projection of the other onto it
  (perpendicularly)
• In this example, a is positive and b is negative.

q
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Dot Product

• If both vectors are normal, the dot product
  defines the cosine of the angle between the
  vectors

In general
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Dot Product

• Note that is q 90 then the dot product 0,
  i.e. the projection of one onto the other has
  zero length ? vectors are orthogonal.
• Also, if q gt 90 then the dot product is negative.

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Cross Product

• Used for defining orientation and constructing
  co-ordinate axes.
• Cross product defined as
• The result is a vector, perpendicular to the
  plane defined by u and v

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Cross Product
Right Handed Coordinate System
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Cross Product

• Cross product is anti-commutative
• It is not associative
• Direction of resulting vector defined by operand
  order

R.H.S.
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Normals Polygons

• Polygons are (usually) planar regions bounded by
  n edges connecting n1 points or vertices.
• For lighting and viewing calculations we need to
  define the normal to a polygon
• The normal distinguishes the front-face from the
  back-face of the polygon.

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Normals Polygons

• First determine the 2 edge vectors from the
  vertices
• The polygon normal is given by

u2
v1
u1
v3
v2