§1.3–Euclidean Space
Tom Lewis
Fall Term 2006
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
1/8
Outline
1
Higher-dimensional real space
2
The Cauchy-Schwarz Inequality
3
The unit cube, ball, and sphere
4
Convexity
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
2/8
Higher-dimensional real space
Elementary terminology
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
3/8
Higher-dimensional real space
Elementary terminology
Hereafter Rm will denote the m-fold Cartesian product of R with
itself. Thus Rm is the collection of ordered m-tuples
x = (x1 , x2 , · · · , xm ),
where xi ∈ R for each 1 ≤ i ≤ m.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
3/8
Higher-dimensional real space
Elementary terminology
Hereafter Rm will denote the m-fold Cartesian product of R with
itself. Thus Rm is the collection of ordered m-tuples
x = (x1 , x2 , · · · , xm ),
where xi ∈ R for each 1 ≤ i ≤ m.
Rm is a vector space over the scalar field of real numbers R with the
usual rules of vector addition and scalar multiplication.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
3/8
Higher-dimensional real space
Elementary terminology
Hereafter Rm will denote the m-fold Cartesian product of R with
itself. Thus Rm is the collection of ordered m-tuples
x = (x1 , x2 , · · · , xm ),
where xi ∈ R for each 1 ≤ i ≤ m.
Rm is a vector space over the scalar field of real numbers R with the
usual rules of vector addition and scalar multiplication.
Given x = (x1 , x2 , · · · , xm ) and y = (y1 , y2 , · · · , ym ) ∈ Rm , the dot
product of x and y is
hx, y i = x1 y1 + x2 y2 + · · · + xm ym .
The dot product of a pair of vectors is a real number.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
3/8
Higher-dimensional real space
Some properties of the dot product
Let x, y , and z be vectors in Rm and let c ∈ R, a scalar. The following
properties are easily verified.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
4/8
Higher-dimensional real space
Some properties of the dot product
Let x, y , and z be vectors in Rm and let c ∈ R, a scalar. The following
properties are easily verified.
Bilinearity hx, y + czi = hx, y i + chx, zi
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
4/8
Higher-dimensional real space
Some properties of the dot product
Let x, y , and z be vectors in Rm and let c ∈ R, a scalar. The following
properties are easily verified.
Bilinearity hx, y + czi = hx, y i + chx, zi
Symmetry hx, y i = hy , xi
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
4/8
Higher-dimensional real space
Some properties of the dot product
Let x, y , and z be vectors in Rm and let c ∈ R, a scalar. The following
properties are easily verified.
Bilinearity hx, y + czi = hx, y i + chx, zi
Symmetry hx, y i = hy , xi
Positive Definiteness hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0, the
zero vector.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
4/8
Higher-dimensional real space
Some properties of the dot product
Let x, y , and z be vectors in Rm and let c ∈ R, a scalar. The following
properties are easily verified.
Bilinearity hx, y + czi = hx, y i + chx, zi
Symmetry hx, y i = hy , xi
Positive Definiteness hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0, the
zero vector.
Definition
The length or magnitude of the vector x = (x1 , x2 , · · · , xm ) is defined to be
q
p
2
|x| = hx, xi = x12 + x22 + · · · + xm
Notice that |x| = 0 if and only if x = 0, the zero vector.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
4/8
The Cauchy-Schwarz Inequality
Inequalities
Our next two results are the Cauchy-Schwarz inequality and the triangle
inequality.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
5/8
The Cauchy-Schwarz Inequality
Inequalities
Our next two results are the Cauchy-Schwarz inequality and the triangle
inequality.
Theorem
For all x, y ∈ Rm , |hx, y i| ≤ |x||y |.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
5/8
The Cauchy-Schwarz Inequality
Inequalities
Our next two results are the Cauchy-Schwarz inequality and the triangle
inequality.
Theorem
For all x, y ∈ Rm , |hx, y i| ≤ |x||y |.
Theorem
For all x, y ∈ Rm , |x + y | ≤ |x| + |y |.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
5/8
The Cauchy-Schwarz Inequality
Inequalities
Our next two results are the Cauchy-Schwarz inequality and the triangle
inequality.
Theorem
For all x, y ∈ Rm , |hx, y i| ≤ |x||y |.
Theorem
For all x, y ∈ Rm , |x + y | ≤ |x| + |y |.
Definition
The Euclidean distance between the vectors x, y ∈ Rm is defined to be the
length of their difference, |x − y |.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
5/8
The unit cube, ball, and sphere
Definition
A box in Rm is a Cartesian product of intervals
[a1 , b1 ] × [a2 , b2 ] × · · · × [am , bm ]
The unit cube is the box
[0, 1]m = [0, 1] × [0, 1] × · · · × [0, 1]
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
6/8
The unit cube, ball, and sphere
Definition
Let y ∈ Rm and let r ≥ 0.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
7/8
The unit cube, ball, and sphere
Definition
Let y ∈ Rm and let r ≥ 0.
The ball of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | ≤ r }
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
7/8
The unit cube, ball, and sphere
Definition
Let y ∈ Rm and let r ≥ 0.
The ball of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | ≤ r }
The unit ball B m of Rm is the ball with radius 1 centered at 0.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
7/8
The unit cube, ball, and sphere
Definition
Let y ∈ Rm and let r ≥ 0.
The ball of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | ≤ r }
The unit ball B m of Rm is the ball with radius 1 centered at 0.
The sphere of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | = r }.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
7/8
The unit cube, ball, and sphere
Definition
Let y ∈ Rm and let r ≥ 0.
The ball of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | ≤ r }
The unit ball B m of Rm is the ball with radius 1 centered at 0.
The sphere of radius r centered at y in Rm is the set
{x ∈ Rm : |x − y | = r }.
The unit sphere S m−1 of Rm is the sphere of radius 1 centered at 0.
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
7/8
Convexity
Definition
Let u, v ∈ Rm . The line segment between u and v is the set of vectors
{x ∈ Rm : x = (1 − t)u + tv for some t ∈ [0, 1]}
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
8/8
Convexity
Definition
Let u, v ∈ Rm . The line segment between u and v is the set of vectors
{x ∈ Rm : x = (1 − t)u + tv for some t ∈ [0, 1]}
Definition
A set E ⊂ Rm is said to be convex provided that for each u, v ∈ E E
contains the line segment between u and v .
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
8/8
Convexity
Definition
Let u, v ∈ Rm . The line segment between u and v is the set of vectors
{x ∈ Rm : x = (1 − t)u + tv for some t ∈ [0, 1]}
Definition
A set E ⊂ Rm is said to be convex provided that for each u, v ∈ E E
contains the line segment between u and v .
Problem
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
8/8
Convexity
Definition
Let u, v ∈ Rm . The line segment between u and v is the set of vectors
{x ∈ Rm : x = (1 − t)u + tv for some t ∈ [0, 1]}
Definition
A set E ⊂ Rm is said to be convex provided that for each u, v ∈ E E
contains the line segment between u and v .
Problem
Is a box convex?
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
8/8
Convexity
Definition
Let u, v ∈ Rm . The line segment between u and v is the set of vectors
{x ∈ Rm : x = (1 − t)u + tv for some t ∈ [0, 1]}
Definition
A set E ⊂ Rm is said to be convex provided that for each u, v ∈ E E
contains the line segment between u and v .
Problem
Is a box convex?
Is a sphere convex?
Tom Lewis ()
§1.3–Euclidean Space
Fall Term 2006
8/8