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Section 4.2: VECTOR SPACES
When you are done with your homework you should be able to…
 Define a vector space and recognize some important vector spaces
 Show that a given set is not a vector space
VECTOR SPACE
A vector space consists of _______________ entities: a _______ of
_____________, a set of _________________, and ___________ operations.
When you refer to a vector space _____, be sure that all four entities are clearly
stated or understood. Unless stated otherwise, assume that the set of scalars is
the set of _______ numbers.
Example 1: Describe the zero vector (the additive identity) of the vector space.
a. C  ,  
b. M 1,4
Example 2: Describe the additive inverse of a vector in the vector space.
a. C  ,   (the set of all realvalued continuous functions
defined on the entire real line.
b. M 1,4
DEFINITION OF A VECTOR SPACE
Let V be a set on which two operations (vector addition and scalar multiplication)
are defined. If the listed axioms are satisfied for every u , v , and w in V and
every scalar (real number) c and d , then V is called a vector space.
Addition
1. u  v is in V .
_____________ under addition
2. u  v  ________________
________________ property
3. u   v  w   ____________
________________ property
4. V has a ___________ vector ____
such that for every ___ in V , ________. additive _________________
5. For every ____ in V , there is a vector in
V denoted by ___ such that _________. additive _________________
Scalar Multiplication
6. cu is in____.
________ under scalar mult.
7. c  u  v   ____________
________________ property
 c  d  u  ____________
________________ property
8.
9. c  du   _______
________________ property
10. 1 u   ____
________________identity
THEOREM 4.4: PROPERTIES OF SCALAR MULTIPLICATION
Let v be any element of a vector space V , and let c be any scalar. Then the
following properties are true.
1. 0 v  ________
2. c 0  ________
3. If ___________, then ___________ or ___________.
4.
 1 v  ________
Example 3: Determine whether the set, together with the indicated operations, is
a vector space. If it is not, then identify at least one of the ten vector space
axioms that fails.
a. The set of all 2 x 2 matrices of the form
a b
c 1


b. The set
 1
 x,
 2


x : x   .


c. The set of all 2 x 2 nonsingular matrices with the standard operations.
Example 4: Rather than use the standard definitions of addition and scalar
3
multiplication in R , suppose these two operations are defined as follows.
a.
 x1 , y1 , z1    x2 , y2 , z2    x1  x2 , y1  y2 , z1  z2 
c  x, y, z    cx, cy, 0 
b.
 x1 , y1 , z1    x2 , y2 , z2    x1  x2  1, y1  y2  1, z1  z2  1
c  x, y, z    cx  c  1, cy  c  1, cz  c  1