Download Lecture 2: Geometry vs Linear Algebra I Points

Lecture 2:
Geometry vs Linear Algebra
Points-Vectors and Distance-Norm
Shang-Hua Teng
2D Geometry: Points
2D Geometry: Cartesian Coordinates
y
(a,b)
x
2D Linear Algebra: Vectors
y
(a,b)
0
x
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
 v1 
 w1 
 v1  w1 
v    and w    then v  w  

v
w
v

w
2
 2
 2
 2
• Scalar Multiplication
 3v1 
  v1 
3v    and  v  

3
v

v
 2
 2
Geometry of Vector Operations
• Vector Addition: v + w
v+w
v
w
Geometry of Vector Operations
• -v
2v
v
-v
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
v-w
w
Norm: Distance to the Origin
• Norm of a vector:
|| v || v1  v2
2
2
 v1 
v 
 v2 
Distance of Between Two Points
dist( v, w) || v  w ||
v1  w1   v2  w2 
2
2
v
v-w
w
Dot-Product (Inner Product)
and Norm
v  w  v1w1  v2 w2
|| v || v  v
Angle Between Two Vectors
v

w
Polar Coordinate
r

v
v  (r cos  , r sin  )  r (cos  , sin  )
Dot Product: Angle and Length
• Cosine Formula
v  rv (cos  , sin  ) and w  rw (cos  , sin  )
v  w  rv rw cos  cos   sin  sin  
 rv rw cos(   )
 ||v||||w| | cos 
v

w
Perpendicular Vectors
• v is perpendicular to w if and only if
vw  0
Vector Inequalities
• Triangle Inequality
|| v  w || ||v||  ||w||
• Schwarz Inequality
| v  w | ||v||||w||
Proof:
| v  w | |v||||w|| | cos  ||| v |||| w ||
3D Points
z
y
x
3D Vector
z
v  (v1 , v2 , v3 )
y
x
Row and Column Representation
v  (v1 , v2 , v3 )
 v1 


v   v2 
v3 
Algebra: Vector Operations
• Vector Addition
 v1 
 w1 
 v1  w1 
v  v2  and w   w2  then v  w  v2  w2 
 
 


v3 
 w3 
v3  w3 
• Scalar Multiplication
 3v1 
  v1 
3v  3v2  and  v   v2 
 


3v3 
 v3 
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
v
u
Norm and Distance
• Norm of a vector:
z
|| v || v1  v2  v3
2
2
2
v  (v1 , v2 , v3 )
y
x
• Distance
dist( v, w) || v  w ||

v1  w1   v2  w2   v3  w3 
2
2
2
Dot-Product (Inner Product)
and Norm
v  w  v1w1  v2 w2  v3 w3
|| v || v  v
v  w  ||v||||w| | cos 
Vector Inequalities
• Triangle Inequality
|| v  w || ||v||  ||w||
• Schwarz Inequality
| v  w | ||v||||w||
Proof:
| v  w | |v||||w|| | cos  ||| v |||| w ||
Dimensions
• One Dimensional Geometry
• Two Dimensional Geometry
• Three Dimensional Geometry
• High Dimensional Geometry
n-Dimensional Vectors and Points
v  (v1 , v2 ,, vn )
 v1 
 v2 
v'   
 
vn 
Transpose of vectors
High Dimensional Geometry
• Vector Addition and Scalar Multiplication
n
• Dot-product v  w   vk wk
k 1
• Norm
|| v || v  v 
n
 vk
2
k 1
• Cosine Formula v  w  ||v||||w| | cos 
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
|| v  w || ||v||  ||w||
• Schwarz Inequality
| v  w | ||v||||w||
Basic Notations
• Unit vector ||v||=1
• v/||v|| is a unit vector
• Row times a column vector = dot product
n
v  w   vk wk  v1 v2
k 1
 w1 
w 
 vn  2 

 
 wn 
Basic Geometric Shapes:
Circles (Spheres), Disks (Balls)
x  c   x  c   r
2
x  c   x  c   r
2