Chapter 4

•Euclidean n-Space

•Linear Transformations from to

•Properties of Linear Transformations to

•Linear Transformations and Polynomials

nR mR

nR mR

DefinitionIf n is a positive integer, then an ordered n-tuple is a sequence of nreal numbers (a1,a2,…,an). The set of all ordered n-tuple is called n-spaceand is denoted by .nR

Note that an ordered n-tuple (a1,a2,…,an) can be viewed either as a “generalized point” or as a “generalized vector”

Definition Two vectors u = (u1 ,u2 ,…,un) and v = (v1 ,v2 ,…, vn) in are calledequal if u1 = v1 ,u2 = v2 , …, un = vn

The sum u + v is defined by

u + v = (u1+v1 , u1+v1 , …, un+vn)

and if k is any scalar, the scalar multiple ku is defined by

ku = (ku1 ,ku2 ,…,kun)

nR

RemarksThe operations of addition and scalar multiplication in this definition are called the standard operations on .nR

If u = (u1 ,u2 ,…,un) is any vector in , then the negative (or additive inverse) of u is denoted by -u and is defined by

-u = (-u1 ,-u2 ,…,-un).

The zero vector in is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0).

The difference of vectors in is defined by

v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)

nR

nR

nR

Theorem 4.1.1 (Properties of Vector in )If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn), and w = (w1 ,w2 ,…,wn) are vectors in and k and m are scalars, then:

a) u + v = v + u

b) u + (v + w) = (u + v) + w

c) u + 0 = 0 + u = u

d) u + (-u) = 0; that is, u – u = 0

e) k(mu) = (km)u

f) k(u + v) = ku + kv

g) (k+m)u = ku+mu

h) 1u = u

nR

DefinitionIf u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn) are vectors in , then theEuclidean inner product u · v is defined by u · v = u1 v1 + u2 v2 +… + un vn

Euclidean Inner Product

ExampleThe Euclidean inner product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in R4 is

u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

nR

It is common to refer to , with the operations of addition, scalar multiplication, and the Euclidean inner product, as Euclidean n-space.

nR

Theorem 4.1.2If u, v and w are vectors in and k is any scalar, then

a) u · v = v · u

b) (u + v) · w = u · w + v · w

c) (k u) · v = k(u · v)

d) v · v ≥ 0; Further, v · v = 0 if and only if v = 0

nR

Example(3u + 2v) · (4u + v)= (3u) · (4u + v) + (2v) · (4u + v )= (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · 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 = (u1 ,u2 ,…,un) in by

Similarly, the Euclidean distance between the points u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) in defined by

nR1/2 2 2 2

1 2|| || ( ) nu u u u u u

nR

2 2 21 1 2 2( , ) || || ( ) ( ) ( )n nd u v u v u v u v u v

ExampleIf u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space 4R

2 2 2 2|| || 1 3 ( 2) 7 63 3 7u

2 2 2 2( , ) || || (1 0) (3 7) ( 2 2) (7 2) 58d u v u v

Theorem 4.1.3 (Cauchy-Schwarz Inequality in )If u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) are vectors in , then |u · v| ≤ || u || || v ||

Theorem 4.1.4 (Properties of Length in )If u and v are vectors in and k is any scalar, then

a) || u || ≥ 0

b) || u || = 0 if and only if u = 0

c) || ku || = | k ||| u ||

d) || u + v || ≤ || u || + || v || (Triangle inequality)

nR

nR

nR

nR

a) d(u, v) ≥ 0

b) d(u, v) = 0 if and only if u = v

c) d(u, v) = d(v, u)

d) d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality)

Theorem 4.1.5 (Properties of Distance in Rn)If u, v, and w are vectors in and k is any scalar, then

Theorem 4.1.6 If u, v, and w are vectors in with the Euclidean inner product, then

2 21 1|| || || ||4 4

u v u v u v

nR

nR

DefinitionTwo vectors u and v in are called orthogonal if u · v = 0

Orthogonality

nR

ExampleIn the Euclidean space the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1)are orthogonal, since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0

Theorem 4.1.7 (Pythagorean Theorem in )If u and v are orthogonal vectors in whith the Euclidean innerproduct, then

4R

nRnR

2 2 2|| || || || || ||u v u v

Alternative Notations for Vectors in nR

It is often useful to write a vector u = (u1 ,u2 ,…,un) in in matrix notation as aRow matrix or a column matrix:

nR

1

21 2[ ]n

n

u

uu or u u u u

u

If we use column matrix notation for the vectors

then

Matrix Formulae for the Dot Product

1 1

2 2

n n

u v

u vu and v

u v

Tu v v u TAu v u A v

Tu Av A u v

A Dot Product View of Matrix Multiplication

Let A = [aij] be an m×r matrix and B =[bij] be an r×n matrix, if the row vectors of A are r1, r2, …, rm and the column vectors of B are c1, c2, …, cn , then the matrix product AB can be expressed as

1 1 1 2 1

2 1 2 2 2

1 1 1

n

n

m m m

r c r c r c

r c r c r cAB

r c r c r c

In particular, a linear system Ax=b an be expressed in dot product form as

1 1

2 2

m m

r x b

r x b

r x b

Where r1, r2,…, rm are the row vectors of A, and b1, b2, …, bm are the entries of b

Example:

1 2 3

1 2 3

1 2 3

3 4 1

2 7 4 5

5 8 0

x x x

x x x

x x x

1 2 3

1 2 3

1 2 3

(3, 4,1) ( , , ) 1

(2, 7, 4) ( , , ) 5

0(1,5, 8) ( , , )

x x x

x x x

x x x