Download Week_1_LinearAlgebra..

GX01 – Robotic Systems Engineering
Dan Stoyanov
George Dwyer, Krittin Patchrachai
2016
Vectors and Vector Spaces
•
•
•
•
Introduction
Vectors and Vector Spaces
Affine Spaces
Euclidean Spaces
2
Vectors
• The most fundamental element in linear algebra is a vector
• Vectors are special types of tuples which satisfy various types
of scaling and addition operations
• We shall predominantly deal with coordinate vectors in this
course but vectors can be more complex, eg. functions
• Vectors actually “live” in a vector space, but we can’t talk about
this until we’ve looked a bit more at the properties of vectors
3
Vectors as Geometric Entities
• Vectors are geometric entities which encode relative
displacement (usually from an origin)
• They do not encode an absolute position
• They are not special cases of a matrix
• They have their own algebra and rules of combination
4
4
Equivalency of Vectors
Equivalent vectors are parallel and of the same length
• Two vectors are the same iff (if and only if):
– They have the same direction
– They have the same length
5
5
Basic Operations on Vectors
• Vectors support addition and scaling:
– Multiplication by a scalar,
– Addition of one vector to another,
6
Scaling Vectors by a Positive Constant
• Length changed
• Direction unchanged
7
7
Scaling Vectors by a Negative Constant
• Length changed
• Direction reversed
8
8
Addition and Subtraction of Vectors
• Summation vectors “closes the triangle”
• This can be used to change direction and length
9
9
Other Properties of Vectors
• Commutativity
• Associativity
• Distributivity of addition over multiplication
• Distributivity of multiplication over addition
10
Vector Spaces
• A vector space
has the following properties:
– Addition and subtraction is defined, and the result is another vector
– The set is closed under linear combinations:
• Given
and
– There is a zero vector
the combination
such that
11
Illustrating Closure and the Zero Vector
12
Examples of Vector and Vector Spaces
Which of the following are vector spaces?
1. A tuple of n real numbers:
2. The zero-vector:
3. The angle displayed on a compass:
13
Answers
• All of the previous examples are vector spaces apart from
an angle on a compass. The reason is that it usually
“wraps around” – e.g., 180 degrees negative becomes 180
degrees positive
14
Yaw Angles Measured by 3 iPhones Over
Time
Angle discontinuities caused by “wrap around”
Summary
• Vectors encode relative displacement, not absolute
position
• Scaling a vector changes its length, but not its direction
• All vectors “live” in a vector space
• We can add scaled versions of vectors together
• The vector space is closed under linear combinations and
possesses a zero vector
• We can analyse the structure of vector spaces looking,
oddly enough, at the problem of specifying the coordinates
of a vector
16
16
Spanning Sets and Vector Spaces
• I am given a vector
• I am given a spanning set of vectors,
• I form the vector w as a linear combination of S,
17
Coordinate Representation of a Vector
• I want to find the set of values l1, ln such that
• What properties must S obey if:
– For any value of x at least one solution exists?
– For any value of x the solution must be unique?
• It actually turns out that it’s easier to answer the second
question first
18
Uniqueness of the Solution
• We want to pick a set S so that the values of l1, ln
are unique for each value of x
• Because everything is linear, it turns out that guaranteeing
linear independence is sufficient
19
Linear Independence
• The spanning set S is linearly independent if
only for the special case
• In other words, we can’t write any member of the set as a
linear combination of other members of the set
20
Dependent Spanning Set Example
• For example suppose that,
where
and a, b are real scalar constants
• This set is not linearly independent and so will yield an
ambiguous solution
21
Linearly Dependent Example
• To see this, compute w from the linear combination of the
vectors of S,
22
Linearly Dependent Example
• Suppose we want to match a vector w which can be
written as
• Matching coefficients, we have three unknowns but just
two equations
23
Linear Independence and Dimensionality
• Therefore, if our basis set
solution is not unique
is linearly dependent, the
• Conversely, the number of linearly independent vectors in
a spanning set defines its dimension
24
Linear Independence in the Dimension of 2
25
Existence of a Solution
• The condition we need to satisfy is that x and w must lie in
the same vector space
• Now,
• Because this is closed under the set of linear
combinations, it must always be the case that
26
Linear Combinations and Vector Spaces
27
Existence of a Solution
• We can go a bit further because we know that
it must be the case that
28
Existence of a Solution
• We have shown that
• Can’t we automatically say that
• No, because what we’ve shown so far is that
29
Arbitrary S Might Not “Fill the Space”
• Therefore, in this case
30
Existence of a Solution
• Therefore, it turns out that a solution is guaranteed to exist
only if
• This means that we must have a “sufficient number” of
vectors in S to point “in all the different directions” in V
• This means that the dimensions of both vector spaces
have to be equal
31
Vector Subspaces
• Consider the set of linearly independent vectors
• Are these spanning sets dependent or independent and
what’s the dimension and basis of the subspace?
32
The Question…
• Recall the question – for any vector
and
where
can we find a unique set of coefficients such that
33
The Answer…
• For S a basis of
we can always represent the vector
uniquely and exactly
• For S a basis of a subspace of
some of the vectors exactly
we can only represent
• Expressing the “closest” approximation of the vector in the
subspace is a kind of projection operation
34
The Answer…
35
Summary
• Vector spaces can be decomposed into a set of
subspaces
• Each subspace is a vector space in its own right (closure;
zero vector)
• The dimension of a subspace is the maximum number of
linearly independent vectors which can be constructed
within that subspace
36
Summary
• A spanning set is an arbitrary set of vectors which
comprise a subspace
• If the spanning set is linearly independent, it’s also known
as a basis for that subspace
• The coordinate representation of a vector in a subspace is
unique with respect to a basis for that subspace
37
37
Changing Basis
• In the last few slides we said we could write the
coordinates of a vector uniquely given a basis set
• However, for a given subspace the choice of a basis is not
unique
• For some classes of problems, we can make the problem
significantly easier by changing the basis to
reparameterise the problem
38
Example of Changing Basis
• One way to represent
position on the Earth is to
use Earth Centered Earth
Fixed (ECEF) Coordinates
• Locally at each point on the
globe, however, it’s more
convenient to use EastNorth-Up (ENU) coordinates
• If we move to a new location,
the ENU basis has to
change in the ECEF frame
Local Tangent Plane
39
Example of an ENU Coordinate System
40
Changing Basis
• Suppose we would like to change our basis set from
to
where both basis span the same vector space
41
Changing Basis
42
Changing Basis
• Since the subspaces are the same, each vector in the
original basis can be written as a linear combination of the
vectors from the new basis,
43
Changing Basis
• Clanking through the algebra, we can repeat this for all the
other vectors in the original basis,
44
Changing Basis
• Now consider the representation of our vector in the
original basis,
• Substituting for just the first coefficient, we get
45
45
Changing Basis
• Substituting for all the other coefficients gives
46
Summary of Changing Basis
• Sometimes changing a basis can make a problem easier
• We can carry this out if our original basis is a subspace of
our new vector space
• The mapping is unique, and corresponds to writing the old
basis in terms of the new basis
• (It’s much neater to do it with matrix multiplication)
47
So What’s the Problem with Vector Spaces?
• We have talked about vector spaces
–
–
–
–
They encode displacement
There are rules for adding vectors together and scaling them
We can define subspaces, dimensions and sets of basis vectors
We can even change our basis
• However vector spaces leave a lot out!
48
The Bits Which Are Missing
• There are no points
– There is no way to represent actual geometric objects
• There is no formal definition of what things like angles, or
lengths mean
– Therefore, we can’t consider issues like orthogonality
• We haven’t discussed the idea of an origin
– Everything floats in “free space”
• Affine spaces start to redress this by throwing points into
the mix
49
More Missing Bits
• We still don’t have a notion of distances
– Just ratios on lines between points
• We still don’t have a notion of angles
• We still don’t have an absolute origin
• These are all introduced in Euclidean geometry
50
Euclidean Spaces
•
•
•
•
Introduction
Tuples
Vectors and Vector Spaces
Euclidean Spaces
51
Euclidean Spaces
• Euclidean spaces extend affine spaces by adding notions
of length and angle
• The Euclidean structure is determined by the forms of the
equations used to calculate these quantities
• We are going to just “give” these without proof
• However, we can motivate the problem by considering the
problem of orthogonally projecting one vector onto another
vector
• First, though, we need some additional vector notation
52
Some More Vector Notation
• Since we are going to define lengths and directions, we
can now decompose a vector into
– A scalar which specifies its length
– An orthonormal vector which defines its direction
Length
(+ve)
Orthonormal vector
(length=1)
53
Projecting Vectors onto Vectors
• Consider two vectors
affine space
and
that occupy the same
• What’s the orthogonal projection of
onto
?
54
The Answer…
Projected vector
55
The Answer…
Projected vector
Projected vector
56
Computing the Answer
• We need to compute both the direction and length of the
projection vector
• The direction of the vector must be parallel to
• Therefore, we are going to define the orthogonal projection
as
Scale
factor
Orthonormal vector
parallel to the “right
direction
57
General Case of the Scalar Product
• If we now let both of our vectors be non-normalised, then
• The scalar product is a special case of an inner product
58
Lengths, Angles and Distances
• Lengths and angles are (circularly) defined as
• The distance function (or metric) between two points is the
length of the vector between them,
59
Properties of Scalar Products
• Bilinearity:
• Positive definiteness:
60
Properties of Scalar Products
• Commutativity:
• Distributivity of the dot product over vector addition,
• Distributivity of vector addition over the dot product,
61
Scalar Product and Projection Mini-Quiz
• What’s the value of
62
Summary
• The Euclidean structure is defined by the scalar product
• The scalar product is used to compute the orthogonal
projection of one vector onto another vector
• The scalar product works in any number of dimensions
• It has many useful properties including bilinearity
• However, it only defines a one dimensional quantity
(length)
• Vector products generalise this
63