Lecture 12: 4.1~4.2 Euclidean Vector Space, Lecture notes of Calculus

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Lecture 12: 4.1~4.

Euclidean Vector Space

Wei-Ta Chu

Definitions  If n is a positive integer, then an ordered n - tuple is a sequence of n real numbers ( a 1 ,a 2 ,…,an ). The set of all ordered n - tuple is called n - space and is denoted by R n .  ( a 1 , a 2 , a 3 ) can be interpreted as a point or a vector in R 3 .  Two vectors u = ( u 1 ,u 2 ,…,un ) and v = ( v 1 ,v 2 ,…,vn ) in R n are called Two vectors u = ( u 1 ,u 2 ,…,un ) and v = ( v 1 ,v 2 ,…,vn ) in R are called equal if u 1 = v 1 ,u 2 = v 2 ,…,un = vn The sum u + v is defined by u + v = ( u 1 + v 1 , u 1 + v 1 ,…,un+vn ) and if k is any scalar, the scalar multiple k u is defined by k u = ( ku 1 ,ku 2 ,…,ku n

Remarks  The operations of addition and scalar multiplication in this definition are called the standard operations on R n .  The zero vector in R n is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0). the vector 0 = (0, 0, …, 0).  If u = ( u 1 ,u 2 ,…,u n ) is any vector in R n , then the negative (or additive inverse) of u is denoted by - u and is defined by - u = (- u 1 ,-u 2 ,…,-u n ).  The difference of vectors in R n is defined by v – u = v + (- u ) = ( v 1

• u 1 ,v 2 - u 2 ,…,v n - u n )

Theorem 4.1.1 (Properties of Vector in R n )  If u = ( u 1 ,u 2 ,…,u n ), v = (v 1 ,v 2 ,…,v n ), and w = ( w 1 ,w 2 ,…, w n ) are vectors in R n and k and l are scalars, then:  u + v = v + u  u + ( v + w ) = ( u + v ) + w u + 0 = 0 + u = u  u + 0 = 0 + u = u  u + (- u ) = 0; that is u – u = 0  k ( l u ) = ( kl ) u  k ( u + v ) = k u + k v  ( k+l ) u = k u + l u  1 u = u

Properties of Euclidean Inner Product  Theorem 4. 1. 2  If u , v and w are vectors in R n and k is any scalar, then  u · v = v · u  ( u + v) · w = u · w + v · w  ( k u) · v = k (u · v)  ( k u) · v = k (u · v)  v · v ≥ 0 ; Further, v · v = 0 if and only if v = 0  Example  ( 3 u + 2 v ) · ( 4 u + v ) = ( 3 u ) · ( 4 u + v ) + ( 2 v ) · ( 4 u + v ) = ( 3 u ) · ( 4 u ) + ( 3 u ) · v + ( 2 v ) · ( 4 u ) + ( 2 v ) · v = 12 ( u · u ) + 11 ( u · v ) + 2 ( v · v )

Norm and Distance in Euclidean n - Space  We define the Euclidean norm (or Euclidean length) of a vector u = ( u 1 ,u 2 ,…,u n ) in R n by  Similarly, the Euclidean distance between the points u = ( u ,u ,…,u ) and v = ( v , v ,…,v ) in R n is defined by 2 2 2 2 1 1 / 2 ( ) ... n u  u  u  u  u   u ( u 1 ,u 2 ,…,u n ) and v = ( v 1 , v 2 ,…,v n ) in R n is defined by  Example  If u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space R 4 2 2 2 2 2 d ( u , v ) u  v  ( u 1  v 1 ) ( u  v ) ...( un  vn ) ( , ) ( 1 0 ) ( 3 7 ) ( 2 2 ) ( 7 2 ) 58 ( 1 ) ( 3 ) ( 2 ) ( 7 ) 63 3 7 2 2 2 2 2 2 2 2               u v u d

Theorems  Theorem 4.1.4 (Properties of Length in R n )  If u and v are vectors in R n and k is any scalar, then  || u || ≥  || u || = 0 if and only if u = 0  |||| kk uu |||| = |= | kk |||||| uu ||||  || u + v ||≤|| u || + || v || (Triangle inequality)

Proof of Theorem 4.1.4 (c)  If u =( u 1 , u 2 ,…, u n ), then k u =( ku 1 , ku 2 ,…, ku n ), so

|| k u || = | k ||| u ||

Theorems  Theorem 4.1.5 (Properties of Distance in R n )  If u , v , and w are vectors in R n and k is any scalar, then  d ( u, v ) ≥  d ( u, v ) = 0 if and only if u = v  d ( u, v ) = d ( v, u )  d ( u, v ) = d ( v, u )  d ( u, v )≤ d ( u, w ) + d ( w, v ) (Triangle inequality)  Proof (d)

Theorems  Theorem 4.1.  If u , v , and w are vectors in R n with the Euclidean inner product, then u · v = ¼ || u + v || 2

• ¼ || u – v || 2  Proof:Proof:

Matrix Formula for the Dot Product

 If we use column matrix notation for the vectors

u = [ u

1

u

2

… u

n

]

T

and v = [ v

1

v

2

… v

n

]

T

or

v

v

u

u

1 1

u and v

then

 n^ n

u v

Matrix Formula for the Dot Product  For vectors in column matrix notation, we have the following formula for the Euclidean inner product: u · v = v T u  For example:

Example

A Dot Product View of Matrix Multiplication  If A = [ aij ] is an m  r matrix and B =[ bij ] is an r  n matrix, then the ij- th entry of AB is ai 1 b 1 j + ai 2 b 2 j + ai 3 b 3 j +… + airbrj which is the dot product of the i th row vector of A and the j th column vector of B column vector of B  Thus, if the row vectors of A are r 1 , r 2 ,…, r m and the column vectors of B are c 1 , c 2 ,…, c n , then the matrix product AB can be expressed as                       1 2 2 1 2 2 2 1 1 1 2 1 m m m n n n AB r c r c r c r c r c r c r c r c r c      