Transcript Document

Recap of linear algebra: vectors,
matrics, transformations, …
Background knowledge for 3DM
Marc van Kreveld
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Vectors, points
• A vector is an ordered pair, triple, … of (real) numbers,
often written as a column
• A vector (3, 4) can be interpreted as the point with
x-coordinate 3 and y-coordinate 4, so (3, 4) as well
• A vector like (2, 1, –4) can be interpreted as a point
in 3-dimensional space
Three times the vector (3, 2),
and the point (3, 2)
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Vectors, length
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Vector addition
• Two vectors of the same dimensionality can be added;
just add the corresponding components:
(a,b) + (c,d) = (a+c, b+d)
• The result is a vector of the same dimensionality
• Geometric interpretation: place one arrow’s start at
the end of the other, and take the resulting arrow
purple + purple = blue
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Scalars, vectors, multiplication
• A value is also called a scalar
• We can multiply a scalar k with a vector (a, b); this is
defined to be the vector (ka, kb)
• Geometric interpretation where a vector is an arrow:
– k = – 1 : reverse the direction of an arrow
– k = 2 : double the length of an arrow; same as adding
a vector to itself
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Vector multiplication
• One type of vector multiplication is called the
dot product, it yields a scalar (a value)
• Example: (a, b, c)  (d, e, f) = ad + be + ef
• It works in all dimensions
• The dot product rule/equality for vectors u and v:
u  v = |u||v| cos 
• Perpendicular vectors have a dot product 0
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Vector multiplication
• Another type of multiplication is the cross product,
denoted by 
• It applies only to two vectors in 3D and yields a
vector in 3D
– the result is normal to the input vectors
– if the input vectors are parallel, we get
the null vector (0, 0, 0)
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Vector multiplication
• The length of the result vector of the cross product
is related to the lengths of the input vectors and
their angle
|a  b| = |a||b| sin 
In words: the length of the
result a  b is the area of the
parallelogram with a and b
as sides
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Vectors
• Other terms of importance:
–
–
–
–
–
linear independence
spanning a (sub)space
basis
orthogonal basis
orthonormal basis
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Matrices
• Matrices are grids of values; an m-by-n (m  n) matrix
consists of m rows and n columns
• An m  n matrix represents a linear transformation
from m-space to n-space, but it could represent many
other things
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Matrices
• A linear transformation:
– maps any point/vector to exactly one point/vector
– maps the origin/null vector to the origin/null vector
– preserves straightness: mapping a line segment (its points)
yields a line segment (its points), which can degenerate to
a single point
Example:
point or vector
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Matrices
mirror in y-axis
shear the x-coordinate
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Matrices
scale x and y by 1.5
rotate by  = /6 radians
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Matrices
• Matrix addition: entry-wise
• Multiplication with scalar: entry-wise
• Multiplication of two matrices A and B:
– #columns of A must match #rows of B
– not commutative
– AB represents the linear
transformation where
B is applied first and A
is applied second
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Matrices
• Other terms of importance:
– identity matrix
– rank of a matrix
– determinant: converts a square matrix to a scalar
Geometric interpretation: tells something about the
area/volume enlargement (2D/3D) of a matrix
Det = 2 (in 2D): a transformed triangle has twice the area
Det = 0: the transformation is a projection
– matrix inversion: represents the transformation that is
the reverse of what the matrix did
– Gaussian elimination: process (algorithm) that allows us
to invert a matrix, or solve a set of linear equations
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Translations and matrices
• A 3x3 matrix can represent any linear transformation
from 3-space to 3-space, but no other transformation
• The most important missing transformation is
translation (which never maps the origin to the origin
so it cannot be a linear transformation)
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Homogeneous coordinates
• Combinations of linear transformations and
translations (one applied after the other) are called
affine transformations
• Using homogeneous coordinates, we can use a 4x4
matrix to represent all 3-dim affine transformations
(generally: (d+1)x(d+1) matrix for d-dim affine tr.)
 the homogeneous coordinates of the point
(a, b, c) are simply (a, b, c, 1)
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Homogeneous coordinates
• The matrix for translation by the vector (a, b, c) using
homogeneous coordinates is:
Just apply this matrix to the origin = (0, 0, 0, 1) and
see where it ends up: (a, b, c, 1)
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Vectors of points
• It is possible to define and use vectors of points:
( (a, b), (c, d), (e,f) ) instead of vectors of scalars
• We can add two of these because vector addition is
naturally defined
• We can also multiply such a thing by a scalar
( (a, b), (c, d), (e,f) ) + ( (g, h), (i, j), (k,l) ) =
( (a, b)+(g, h), (c, d)+(i, j), (e,f)+(k,l) ) =
( (a+g, b+h), (c+i, d+j), (e+k, f+l) )
3 ( (a, b), (c, d), (e,f) ) = ( 3(a, b), 3(c, d), 3(e,f) ) =
( (3a, 3b), (3c, 3d), (3e, 3f) )
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Vectors of points
• We can not add such a thing and a normal 3D vector
because we cannot add a scalar and a vector/point
( (a, b), (c, d), (e,f) ) + ( g, h, i ) = undefined
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Vectors of points
• We can even apply (scalar) matrices to these things:
This works be cause we know how to add points
and multiply scalars and points
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Questions
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Questions
5. Let S be the collection of all strings. Define
– addition of two strings as their concatenation
– multiplication of a string with a nonnegative integer by repeating the
string as often as the value of the integer
Compute:
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