Math 260 Linear Algebra Handout 3 - Dr. McLoughlin’s Class page 1 of 6
MATH 260 LINEAR ALGEBRA
DR. MCLOUGHLIN’S CLASS
EUCLIDEAN VECTOR SPACE DEFINITIONS, LEMMAS, THEOREMS,
AND COROLLARIES
HANDOUT 3
Let U = n where n    n > 1.
Definition 1: Let x , y , and z be a matrices of size 1  n where n    n > 1.
We call x , y , and z vectors of size 1  n (or just simply vectors) and we say x , y , z  n .
x = (x1, , x n ) , y = (y1, , yn ) , and z = (z1, , z n ) .
Definition 2-2: For 2 denote the unit vectors (1, 0) as m , and (0, 1) as n .
Definition 2-3: For 3 denote the unit vectors (1, 0, 0) as i , (0, 1, 0) as j , and (0, 0, 1) as k .
Theorem 1-1: Let x  2 It is the case that  p, q    x = p m + q n
Theorem 1-2: Let x  3 It is the case that  a, b, c    x = a i + b j + c k .
Definition 3: Let x , y  n and p  
x + y = (x1  y1, , x n  y n )
px = (p  x1,
, p  xn )
n
n
Definition 4: Let x , y 
xy=
 x i yi
i 1
n
Theorem 2: Let x , y , z 
xx  0
xy= yx
(x  y)  z = (x  z)  (y  z)
(px)  y = p(x  y)
xx  0  x  0
 p   It is the case that
n
Definition 5: Let x 
n
x = (x  x)
0.5
=
 xi2
i 1
n
Definition 6: Let x , y 
d( x , y ) = x  y
Theorem 3 (Cauchy – Schwartz): Let x , y 
Theorem 4: Let x , y 
n
n
 p  . It is the case that
It is the case that x  y  x  y
Math 260 Linear Algebra Handout 1 - Dr. McLoughlin’s Class page 2 of 2
x 0
x 0  x0
px  p x
xy  x  y
Theorem 5: Let x , y  n  x  y is well defined.
2 1
2
1
xy 
xy
It is the case that x  y =
4
4


n
Definition 7: Let x , y 
Theorem 6: Let x , y 
n
 x  y is well defined whilst x  y are orthogonal.
 xy   x   y 
Theorem 7: Let x , y 
It is the case that
2
n
2
 x  y is well defined
x  y = yT x when x , y are considered vectors and . x and y are considered matrices.
Claim 1: Let x , y 
It is the case that

x  y = 0  x  y are orthogonal.
2
It is the case that

n
 x  y is well defined and let A be a matrix of size n  n
 xA  y  x   yAT  
Last revised 23 October 2005
 
 
x  yA  xAT  y
© 2000, 2001 – 2005 M. P. M. M. M.