1. Linear Algebra
Chapter 4
Vector Spaces

2. 4.1 General Vector Spaces Definition
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
u + v…………… (V is closed under addition.)
cu …………… (V is closed under scalar multiplication.)

3. 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 = ……
called the ……… of u, such that u + (-u) = 0.
Scalar Multiplication Axioms
7. c(u + v) = ……………
8. (c + d)u = ……………
c(du) = ……………
1u = ……………

4. A Vector Space in R3 Is V a vector space ? Is Z a vector space ?
Example 1
Is V a vector space ?
Solution
Example 2
Is Z a vector space ?
Solution
Ch04_4

5. Prove that W is a vector space.
Example 3
Prove that W is a vector space.
Proof

6. Vector Spaces of Matrices (Mmn)
Prove that M22 is a vector space.
Proof

7. In general: The set of m  n matrices, Mmn, is a vector space.

8. Example 4
Solution
Ch04_8

9. Vector Spaces of Functions
Prove that F = { f | f : R R } is a vector space.

10. Vector Spaces of Functions (continued)

11. Vector Spaces of Functions (continued)
Example 5
Is the set F ={ f | f (x)=ax2+bx+c , a,b,c R , } a vector space?
Solution
Ch04_11

12. 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:

13. 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.

14. 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

15. Example 8
Prove that the set W of 2  2 diagonal matrices is a subspace of the vector space M22.
Solution
Ch04_15

16. 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