Vector Spaces

Question 1

S is closed under addition if…

Answer 1

s1 + s2 is in S

Question 2

S is closed under scalar multiplication if…

Answer 2

a.s is in S where a is in the field

Question 3

Axioms of vector space

Answer 3

V1: u + v = v + u
V2: (u+v)+w = u+(v+w)
V3: v + 0  = v
V4: v + (-v) = 0 
V5: α(u+v) = αu + αv
V6: (α + β)v = αv + βv
V7: (αβ)v = α(βv)
V8: 1v = v

Question 4

Basic properties of a vector space

Answer 4

α0 = 0
0v = 0
αv = 0, then either α = 0 or v = 0
(-α)v = α(-v)

Question 5

Define a subspace

Answer 5

Let V be a vector space over F. A subspace W of V is a non-empty subset of V which itself forms a vector space under the same operations.

Question 6

Let V be a vector space over a field F. Let W be a non-empty subset of V.
Then W is a subspace of V if and only if…

Answer 6

• W is closed under addition

- W is closed under scalar multiplication

Question 7

For S and T subspaces of V….
S n T
S U T

Answer 7

S n T is a subspace of V

S U T is not necessarily a subspace of V