Lecture 2:Geometry vs Linear Algebra

Points-Vectors and Distance-Norm

Shang-Hua Teng

2D Geometry: Points

2D Geometry: Cartesian Coordinates

x

y

(a,b)

2D Linear Algebra: Vectors

x

y

(a,b)

0

2D Geometry and Linear Algebra

• Points

• Cartesian Coordinates

• Vectors

2D Geometry: Distance

2D Geometry: Distance

How to express distance algebraically using coordinates???

Algebra: Vector Operations

• Vector Addition

• Scalar Multiplication

22

11

2

1

2

1 then and wv

wvwv

w

ww

v

vv

2

1

2

1 and 3

33

v

vv

v

vv

Geometry of Vector Operations

• Vector Addition: v + w

v

w

v + w

Geometry of Vector Operations

• -v

v

-v

2v

Linear Combination

Linear combination of v and w

{cv + d w : c, d are real numbers}

Geometry of Vector Operations

• Vector Subtraction: v - w

v

w

v + w

v - w

Norm: Distance to the Origin

• Norm of a vector: 22

21|||| vvv

2

1

v

vv

Distance of Between Two Points

v

wv - w

222

211||||),dist( wvwvwvwv

Dot-Product (Inner Product)and Norm

2211 wvwvwv

vvv ||||

Angle Between Two Vectors

v

w

Polar Coordinate

vr

)sin,(cos)sin,cos( rrrv

Dot Product: Angle and Length

• Cosine Formula

cos

)cos(

sinsincoscoswv

)sin,(cos and )sin,(cos

| ||v||||w|

rr

rr

rwrv

wv

wv

wv

v

w

Perpendicular Vectors

• v is perpendicular to w if and only if

0wv

Vector Inequalities

• Triangle Inequality

• Schwarz Inequality

||w||||v||wv ||||

||v||||w||wv ||

|||||||||cos||| wv|v||||w||wv Proof:

3D Points

y

x

z

3D Vector

y

x

z

),,( 321 vvvv

Row and Column Representation

),,( 321 vvvv

3

2

1

v

v

v

v

Algebra: Vector Operations

• Vector Addition

• Scalar Multiplication

33

22

11

3

2

1

3

2

1

then and wv

wvwv

wvw

ww

wv

vv

v

3

2

1

3

2

1

and 3

33

3v

vv

vv

vv

v

Linear Combination

• Linear combination of v (line)

{cv : c is a real number}

• Linear combination of v and w (plane)

{cv + d w : c, d are real numbers}

• Linear combination of u, v and w (3 Space)

{bu +cv + d w : b, c, d are real numbers}

Geometry of Linear Combination

u

u

v

Norm and Distance• Norm of a vector:

• Distance

23

22

21|||| vvvv

y

x

z

),,( 321 vvvv

233

222

211

||||),dist(

wvwvwv

wvwv

Dot-Product (Inner Product)and Norm

332211 wvwvwvwv

vvv ||||

cos| ||v||||w|wv

Vector Inequalities

• Triangle Inequality

• Schwarz Inequality

||w||||v||wv ||||

||v||||w||wv ||

|||||||||cos||| wv|v||||w||wv Proof:

Dimensions

• One Dimensional Geometry

• Two Dimensional Geometry

• Three Dimensional Geometry

• High Dimensional Geometry

n-Dimensional Vectors and Points

),,,( 21 nvvvv

nv

vv

v

2

1

'

Transpose of vectors

High Dimensional Geometry

• Vector Addition and Scalar Multiplication

• Dot-product

• Norm

• Cosine Formula

n

kkk wvwv

1

n

kkvvvv

1

2||||

cos| ||v||||w|wv

High Dimensional Linear Combination

• Linear combination of v1 (line)

{c v1 : c is a real number}

• Linear combination of v1 and v2 (plane)

{c1 v1 + c2 v2 : c1 ,c2 are real numbers}

• Linear combination of d vectors v1 , v2 ,…, vd

(d Space)

{c1v1 +c2v2+…+ cdvd : c1,c2 ,…,cd are real numbers}

High Dimensional Algebra and Geometry

• Triangle Inequality

• Schwarz Inequality

||w||||v||wv ||||

||v||||w||wv ||

Basic Notations

• Unit vector ||v||=1

• v/||v|| is a unit vector

• Row times a column vector = dot product

n

n

n

kkk

w

w

w

vvvwvwv

2

1

211

Basic Geometric Shapes:Circles (Spheres), Disks (Balls)

2

2

rcxcx

rcxcx