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Linear Algebra
Chapter 4
Vector Spaces
4.1 General Vector Spaces
Our aim in this section will be to focus on the algebraic
properties of Rn.
Definition
A ……………… is a set V of elements called vectors, having
operations of addition and scalar multiplication defined on it
that satisfy the following conditions:
Let u, v, and w be arbitrary elements of V, and c and d are scalars.
• Closure Axioms
1. u + v……………
(V is closed under addition.)
2. cu ……………
(V is closed under scalar multiplication.)
Ch04_2
Definition of Vector Space (continued)
•
Addition Axioms
3. u + v = ……………
(commutative property)
4. u + (v + w) = …………… (associative property)
5. There exists an element of V, called the …………, denoted 0,
such that u + 0 = ……
6. u V  - u called the ……… of u, such that u + (-u) = 0.
•
Scalar Multiplication Axioms
7. c(u + v) = ……………
8. (c + d)u = ……………
9. c(du) = ……………
10. 1u = ……………
Ch04_3
A Vector Space in R3
Example 1
Let V  {, -3, -1, 1, 3, 5, 7,}. Is V a vector space ?
Solution
Example 2
Let Z  {, -2, -1, 0, 1, 2, 3, 4,}. Is Z a vector space ?
Solution
Ch04_4
Example 3
Let W  { a(1, 0, 1) | a  R }. Prove that W is a vector space.
Proof
Ch04_5
Vector Spaces of Matrices (Mmn)
 p q
Let M 22  { 
| p, q, r , s  R }. Prove that M22 is a vector space.

 r s
Proof
Ch04_6
In general: The set of m  n matrices, Mmn, is a vector space.
Ch04_7
Example 4
p
Is the set W  { 
r
q
| p,q,r ,s  0} a vector space?

s
Solution
Ch04_8
Vector Spaces of Functions
Prove that F = { f | f : R R } is a vector space.
Ch04_9
Vector Spaces of Functions (continued)
Ch04_10
Vector Spaces of Functions (continued)
Example 5
Is the set F ={ f | f (x)=ax2+bx+c , a,b,c R , a  0 } a vector space?
Solution
Ch04_11
Subspaces
Definition
Let V be a vector space and U be a …………………………. of V.
U is said to be a …………… of V if it is
……………………….. and …………………………………..
Note:
........

........ is a vector space
........

Ch04_12
Example 6
Let U be the subset of R3 consisting of all vectors of the form
(a, a, b) , a,bR , i.e., U = {(a, a, b)  R3 }.
Show that U is a subspace of R3.
Solution
Show that U = {(a, 0, 0)  R3 , a R } is a subspace of R3.
Ch04_13
Example 7
Let V be the set of vectors of of R3 of the form (a, a2, b),
V = {(a, a2, b)  R3 , a,b R }. Is V a subspace of R3 ?
Solution
Ch04_14
Example 8
Prove that the set W of 2  2 diagonal matrices is a subspace of
the vector space M22.
Solution
Ch04_15
Theorem 4.5 (Very important condition)
Let U be a subspace of a vector space V.
………………………………………
Example 9
Let W be the set of vectors of the form (a, a, a+2).
Show that W is not a subspace of R3.
Solution
Ch04_16