Math 211- Midterm 2 Flashcards
Linearity Condition
L((s ⊙V x) ⊕V (t ⊙V y)) = (s ⊙W L(x)) ⊕W (t ⊙W L(y))
Span of polynomials
Span(B) = {t1p1(x) + · · · + tkpk(x) | t1, …, tk ∈ R}
Linear Independence of Span
The set B = {p1(x), …, pk(x)} is said to be linearly independent if the only solution to the equation
t1p1(x) + · · · + tkpk(x) = 0
is t1 = · · · = tk = 0; otherwise, there is a solution where not all ti are zero, for which B is said to be linearly
dependent.
Monomial Basis
The collection of polynomials of degree at most n is denoted Pn. The collection of polynomials given by
{1, x, x2, …, xn}
Closure under add.
x ⊕ y ∈ V
Commutativity of add
x ⊕ y = y ⊕ x
Associativity of add.
(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)
Existence of zero
There is a 0 ∈ V such that 0 ⊕ v = v for all v ∈ V
Existence of inverse
For each v ∈ V there exists a −v ∈ V such that v ⊕ −v = 0
Closure under mult.
s ⊙ x ∈ V
Associativity of mult.
s ⊙ (t ⊙ x) = (st) ⊙ x
Distributivity of scalar add.
(s + t) ⊙ x = (s ⊙ x) ⊕ (t ⊙ x)
Distributivity of add.
t ⊙ (x ⊕ y) = (t ⊙ x) ⊕ (t ⊙ y)
1 is the scalar identity
1 ⊙ v = v for all v ∈ V
10 Axioms of vector space
V1 Closure under add.
V2 Commutativity of add.
V3 Associativity of add.
V4 Existence of zero
V5 Additive inverse
V6 Closure under scalar multi.
V7 Associativity of scalar multi.
V8 Distributivity of scalar add.
V9 1 is the scalar identity
The following are vector spaces:
The following are vector spaces:
1. The space Rn equipped with vector addition and scalar multiplication.
2. The space M(m, n) of all size m×n matrices equipped with matrix addition and scalar multiplication.
3. The space Pn equipped with polynomial addition and scalar multiplication.
subspace of V if it satisfies the following properties:
- Non Empty
- Closure under addition
- Closure under scalar multiplication
subspace of a vector space
is a vector space itself. Consequently, this means that as U ⊆ V you may think of a vector subspace as a smaller vector space within another vector space.
