Download Introductory Notes on Vector Spaces

CIS 457 Computer Graphics
Introduction to Vector Spaces
Definition A set V (set of vectors) is a vector space over a field F (set of scalars) provided:
1. V is an abelian group (operation denoted as +)
2. An operation, referred to as scalar multiplication, with left operand coming from F and right
operand coming from V, such that for all a, b ε V and for all α, β ε F :
i) αv ε V
ii) α(βa) = (αβ)a
iii) (α+β)a = αa + βa
iv) α(a + b) = αa + αb
v) 1a = a
Note that an abelian group has the property of commutativity, so that for any two vectors, v1 and
v2, v1 + v2 = v2 + v1. Also note that a field has two operations defined on it (usually denoted +
(plus) and . (or simply juxtaposition) (times)), so that for any two scalars α and β, α + β and αβ
are scalars.
Also note, that from the above properties, it can be shown that 0v = 0 (the 0 vector) (note here
that the 0 on the left of the equal sign is a scalar (the additive identity for the field F, and the 0 on
the right of the equal sign is the 0 vector in V (the identity element for the abelian group V)).
Proof: 0v = (0 + 0)v = 0v + 0v; now cancel 0v from both sides (cancellation laws valid in a
group), obtaining 0 (vector) = 0v.
Important: In a vector space, a scalar times a vector is a vector (see i) above).
Other Definitions related to vector spaces.
1. A linear combination of vectors v1, v2, …, vn is a sum of the form:
α1 v1 + α2 v2 + … + αn vn where αi, i = 1…n, are scalars.
2. A set of vectors { v1, v2, …, vn }, a subset of V, spans the vector space V, provided each
vector in V is a linear combination of v1, v2, …, vn. If a finite set of vectors spans the
vector space, the vector space is said to be finitely generated.
3. The vectors v1, v2, …, vn are linearly independent provided
α1 v1 + α2 v2 + … + αn vn = 0 (vector) implies αi = 0 for all i = 1, …, n.
4. A set of vectors B = { v1, v2, …, vn } is a basis for the vector space V provided
i) B spans the vector space V
and
ii) v1, v2, …, vn are linearly independent
Vector spaces used in Computer Graphics
R2 = { (x, y) | x ε R and y ε R} (R is the set of real numbers)
R3 = {(x, y, z) | x ε R, y ε R, z ε R}
For both of these, the field (set of scalars) over which they are vector spaces is R,
the set of real numbers.
Addition of vectors in R2 and R3 is done component-wise.
E.g.s.:
(2, -5) + (4.1, 11.89) = (6.1, 6.89)
(π, 1.29, - 18) + (2, 8, -5.6) = (π + 2, 9.29, -23.6)
Scalar multiplication in R2 and R3 is performed by multiplying each
component of the vector by the scalar.
E.g.s.:
5.1(2, -5) = (10.2, -25.5)
3(π, 1.29, - 18) = (3π, 3. 87, -54)
Note that {(1, 0), (0,1)} is a basis for R2. It is easy to prove this (see class notes).
Note that {(1,0,0), (0,1,0), (0,0,1)} is a basis for R3. It is easy to prove this.
In R3, names for the above basis vectors are:
i = (1, 0, 0)
j = (0, 1, 0) k = (0, 0, 1)
Important: In R2 and R3, vectors also have length and direction. Geometrically,
it is often helpful to view a vector (x, y) in R2 as an arrow originating at the origin
in the Cartesian coordinate plane (the point (0,0)), and terminating (with arrow
head) at the point (x, y). Similarly, a vector (x, y, z) in R3 can be viewed as an
arrow whose tail is at the origin of Cartesian 3-space (the point (0,0,0)), and whose
arrow head is at the point (x, y, z).
The length of (x, y) in R2, denoted |(x,y)|, is defined to be the square root of ( x
squared plus y squared), i.e., it is the distance from the point (x, y) to the origin
(0,0).
The length of (x, y, z) in R3, denoted |(x, y, z)|, is defined to be the square root of
(x squared plus y squared plus z squared), i.e., it is the distance from the point (x,
y, z) to the origin (0,0,0).
The directions of vectors in R2 and R3 are given by the angles made by the arrow
representations of the vectors and the coordinate axes. In R2, one can simply think
of the direction of a vector as being the slope of the line through the origin and the
point (x, y). This slope would simply be y/x.
Definition: A unit vector in R2 or R3 is a vector of length 1.
Important notes about vectors in R2 and R3:
1. Multiplying a vector v by a scalar results in a vector having the same direction
as v, if the scalar is positive, or in a vector having the opposite direction as v,
if the scalar is negative. If the scalar is 0, the resulting vector is the 0 vector.
Example: 10(2,3) = (20, 30), a vector 10 times as long as (2,3) but in the
same direction as (2,3).
-10(2, 3) = (-20, -30), a vector 10 times as long as (2,3) but in the
opposite direction as (2,3).
0(2, 3) = (0, 0), the 0 vector.
2. Dividing a vector v by its length (i.e. multiplying v by 1/|v|) gives a unit
vector in direction of v.
Example: |(3,4)| = 5
1/5(3,4) = (3/5, 4/5) a unit vector in direction of (3,4).
Also note that since (3/5, 4/5) is a unit vector in the direction of (3, 4),
to get a vector of length L (L >= 0) in same direction as (3, 4) multiply the
vector (3/5, 4/5) by L. Example: 15(3/5, 4/5) = (9, 20) is a vector of length 15
in the same direction as (3, 4).
3. In R2, vectors perpendicular to the vector (x, y) are (-y, x) and (y, -x). Recall
that two lines are perpendicular if their slopes are negative reciprocals of each
other. The direction of the vector (x,y) (i.e. the slope of line through origin and
point (x,y)) = y/x. The directions of the vectors (slopes of lines) of (-y, x) and
(y, -x) are (-x, y).
Example : the vectors (-4, 3) and (4, -3) are perpendicular to (and have same
length as) the vector (3, 4).