Math 260 Linear Algebra Handout 4 - Dr. McLoughlin’s Class page 1 of 4
MATH 260 LINEAR ALGEBRA
DR. MCLOUGHLIN’S CLASS
GENERAL VECTOR SPACE
DEFINITIONS, LEMMAS, THEOREMS, AND COROLLARIES
HANDOUT 4
Theorem 5-1: Let V be a vector space with + and . well-defined (e.g.:
V, ,  is a well defined
vector space. Let x  V and c   It is the case that
1)
0x = 0
2)
c0 = 0
3)
(–c) x =  x
4)
c x = 0  (c = 0  x = 0 )
Definition 5-1: Let V, ,  be a well-defined vector space. Let W  V. If W, ,  is a vector
space itself, then W is called a subspace of V.
Theorem 5-2: Let V, ,  be a well-defined vector space. Let W  V. W is a subspace of V if
and only if
1) W is closed under +; and,
2) scalar multiplication for W is closed.
Theorem 5-3: Let us consider
n
, ,  . Let us consider A x = 0 be a system of m equations
with n unknowns. The set of solutions, S, to A x = 0 is a subspace of
n
(e.g.: S, ,  is a
vector space).
Theorem 5-4: Let V, ,  be a well-defined vector space. Let W  V  |W| = n where n  .
Without loss of generality let W = { w1 , w 2 , . . . , w n }.
n
C = {x| x=
a w
i 1
i
i
where ai  , i  n}is a subspace of V and moreover
C is the ‘smallest’ subspace of V that contains w1 , w 2 , . . . , w n (e.g.: If V1  V  V1 , ,  is a
vector space containing w1 , w 2 , . . . ,  w n , then W  V1).
Definition 5-2: Let V, ,  be a well-defined vector space. Let W  V. If W, ,  is a vector
space itself, then W is called a subspace of V. Let S = { s1 , s2 , . . . , s j } be a set of vectors in V.
Math 260 Linear Algebra Handout 4 - Dr. McLoughlin’s Class page 2 of 4
j
The set P = { x | x =
a s
i 1
i i
where ai  , i  j}is a subspace of V with the inherited + and .
from V and P is called the span of S or more simply P = span(S).
Theorem 5-5 Let V, ,  be a well-defined vector space.
Let W ={ w1 , w 2 , . . . , w n }  C = span(W) = { x | x =
n
a w
i 1
and S = { s1 , s2 , . . . , s j }  P = span(S) = { y | y =
j
b s
i 1
i i
i
i
where ai  , i  n}
where bi  , i  j}
span(S) = span(W) iff  y  P y is a linear combination of the vectors in W and

x  C x is a linear combination of the vectors in S.
Theorem 5-61 Let V, ,  be a well-defined vector space.
Let W ={ w1 , w 2 , . . . , w n }  C = span(W) = { x | x =
n
a w
i 1
i
i
where ai  , i  n}
 w1 
 
w2 
 . 
Let A =   (the matrix created by each row being on of the vectors in W)
 . 
 . 
 
 w n 
y  span(W) iff det( A )  0
Corollary 5-62 Let V, ,  be a well-defined vector space. Let y  V and
W ={ w1 , w 2 , . . . , w n }
y  span(W) when det( A ) = 0 for A defined in theorem 5-6.
Definition 5-3 Let V, ,  be a well-defined vector space.
1
A consequence of theorem 4.3.4 part (e0 and part (g) , page 206 of Elementary Linear Algebra, Anton & Rorres, 9yh
Ed.
2
This is a codification of the results discussed in class as to Example 12, page 237.
Math 260 Linear Algebra Handout 4 - Dr. McLoughlin’s Class page 3 of 4
Let W ={ w1 , w 2 , . . . , w n }  consider the equation
n
a w
i 1
i
i
=0
where ai  , i  n
If a1  a 2   a n  0 is the only solution then W is a set of linearly independent vectors; whilst,
if a1  a 2   a n  0 is not the only solution then W is not a set of linearly independent vectors
n
and we say they are linearly dependent
(e.g.:  j  n  aj  0 for a solution of  a i w i
= 0 ).
i 1
Theorem 5-7 Let V, ,  be a well-defined vector space. Let W be a set of vectors such that
|W|  2. It is the case that the vectors are linearly dependent iff  at least one vector that is a linear
combination of the other vectors in W
Theorem 5-8 Let V, ,  be a well-defined vector space. Let W be a finite set of vectors such that
0  W. It is the case that the vectors are linearly dependent.
Theorem 5-9: Let us consider
n
, ,  where + and . are the usual operations.
Let W be a set of vectors such that |W| > n. It is the case that the vectors are linearly dependent.
Theorem 5-10: Let us consider
n
, ,  where + and . are the usual operations.
Let W be a set of vectors such that |W| < n. It is the case that the vectors do not span
n
.
Definition 5-4 Let V, ,  be a well-defined vector space.
Let B ={ b1 , b 2 , . . . , bn } . B is a basis for V iff
B is a set of linearly independent vectors and B spans V.
Theorem 5-11: Let V, ,  be a well-defined vector space and B ={ b1 , b 2 , . . . , bn }be a basis
for V. Let v  V.  a unique set of real numbers r1 , r2 , . . . , rn  v =
n
rb
i 1
i
i
Definition 5-5 Let V, ,  be a well-defined vector space.
V is called a finite-dimensional (in a Linear Algebra sense) vector space iff  a basis for V, say B,
such that | B | = n where n  
V is called an infinite-dimensional (in a Linear Algebra sense) vector space iff no such basis exists
for V.
Theorem 5-12: Let V, ,  be a well-defined finite-dimensional vector space and let
K ={ k1 , k 2 , . . . , k m }be a basis for V (m  ).
Let J be a set of vectors from V such that | J | = n where n    n > m. The vectors in J are
linearly dependent.
Math 260 Linear Algebra Handout 4 - Dr. McLoughlin’s Class page 4 of 4
Let J be a set of vectors from V such that | J | = n where n    n < m. The vectors in J do not
span V.
Important note 5-12: Let V, ,  be a well-defined finite-dimensional vector space and let
K ={ k1 , k 2 , . . . , k m }be a basis for V (m  ).
Let J be a set of vectors from V such that | J | = n where n    n = m.
The vectors in J may or may not be linearly dependent whilst indeed the vectors in J may or may
not span V.
Theorem 5-13: Let V, ,  be a well-defined finite-dimensional vector space and let
m be the dimension for V (m  ) (e.g.: dim(V) = m).
Let W  V and let W, ,  be a vector space itself (W is a subspace of V).
dim(W)  dim(V)
Theorem 5-14: Let V, ,  be a well-defined finite-dimensional vector space and let
m be the dimension for V (m  ) (e.g.: dim(V) = m).
Let W  V and let W, ,  be a vector space itself (W is a subspace of V).
dim(W) = dim(V)  W = V.
Last revised 23 November 2005
© 2000, 2001 – 2005 M. P. M. M. M.