Lesson 11
MA26200, Gabriel Sosa
July 7, 2014 - Monday
SECTION 4.2. VECTOR SPACES
By the end of this lesson you should be familiar with the following concepts:
1. Vector Space over the Real (or Complex) Field
2. Sets closed under addition
3. Sets closed under scalar multiplication
4. The Commutative and Associative properties for addition
5. Identity element for addition (Zero Vector in a Vector Space) and Additive inverses.
6. The Unit property for scalar multiplication.
7. The Associative property of scalar multiplication.
8. The Distributive properties of scalar multiplication over vector addition and scalar addition.
You should also be familiar with the notation:
Rn , for the standard vector space of all ordered n-tuples of real numbers with the standard
vector addition and scalar multiplication.
Mm×n (R) for the vector space of all m × n matrices with entries in the real numbers
with the standard matrix addition and scalar multiplication of a matrix. (If the matrices
are square we simply write Mn (R)).
Pn for the vector space of all polynomials with coefficients in the real numbers of degree
at most n.
C k (I) for the vector space of all real functions with at least k continuous derivatives on
the interval I.
Additionally you should be able to:
1. Verify if a defined addition on a set is closed, commutative, associative, possesses an
identity element and/or additive inverses.
2. Verify if a defined scalar multiplication is closed, associative, possesses the unit property
and/or distributes over the defined addition and/or scalar addition.
3. Recall that in a vector space:
(a) The zero vector is unique and does not depend on the element it is being added
to.
→
−
→
−
→
−
−
−
−
−
−
(b) 0 · →
v = 0, 1·→
v =→
v , k · 0 = 0 and (−1) · →
v = −→
v
→
−
→
−
−
−
(c) If k · →
v = 0 , then k = 0 or →
v = 0.
Lesson 11
MA26200 (Gabriel Sosa), Summer ’14
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SETS CLOSED UNDER ADDITION.
1. Consider the set of even numbers E = {. . . , −4, −2, 0, 2, 4, . . . }, is it closed under
standard addition?
2. Consider the set of odd numbers O = {. . . , −3, −1, 1, 3, . . . }, is it closed under standard addition?
3. Consider the set of 2 × 3 matrices, M2×3 , and define the addition of two matrices in
the following way A ⊕ B = AB T , is this set closed under this definition of addition?
SETS CLOSED UNDER SCALAR MULTIPLICATION
1. Consider the set of all real numbers R, and define a scalar multiplication as
k x = xk+1 , is this set closed under scalar multiplication?
2. Consider the set of all functions with continuous derivatives in the real line, C 1 (R),
and define a scalar multiplication as k f (x) = f (kx), is this set closed under scalar
multiplication?
3. Consider the set of all polynomials with real coefficients of degree at most 2, P2 , and
p(kx) + kp(x)
, is this set closed under scalar
define a scalar multiplication as k p(x) =
2
multiplication?
Lesson 11
MA26200 (Gabriel Sosa), Summer ’14
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COMMUTATIVE PROPERTY OF ADDITION. x ⊕ y = y ⊕ x
ASSOCIATIVE PROPERTY OF ADDITION. (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)
IDENTITY ELEMENT FOR ADDITION. (ZERO VECTOR ON A VECTOR SPACE).
ADDITIVE INVERSES. For every a there is a e
a such that a ⊕ e
a=e
a ⊕ a = e, the identity
element/zero vector.
1. Define addition for 2 × 2 matrices in the following way A ⊕ B = AT + B T .
2. Consider the set R, and define x ⊕ y = xy + 2x + 2y + 2.
3. Define addition in R2 in the following way:
(x1 , y1 ) ⊕ (x2 , y2 ) = (x1 + x2 , y1 y2 ).
4. Define addition for functions with continuous derivative in the interval [0, 1], C 1 ([0, 1])
in the following way f (x) ⊕ g(x) = f (g(x)).
UNIT PROPERTY OF THE SCALAR MULTIPLICATION. 1 x = x.
ASSOCIATIVE PROPERTY OF THE SCALAR MULTIPLICATION. (k1 k2 )x = k1 (k2 x).
DISTRIBUTIVE PROPERTY OF THE SCALAR MULTIPLICATION OVER SET ADDITION.
k (x ⊕ y) = (k x) ⊕ (k y).
DISTRIBUTIVE PROPERTY OF THE SCALAR MULTIPLICATION OVER SCALAR ADDITION. (k1 + k2 ) x = (k1 x) ⊕ (k2 x).
1, Consider the following addition and scalar multiplication in R2 ,
(x1 , x2 ) ⊕ (y1 , y2 ) = (x1 , y2 ) and k (x1 , y1 ) = (x1 , ky1 ).
2. Consider the following addition and scalar multiplication in M3 (R):
A ⊕ B = A + B − I and k A = kA + (1 − k)I.
Lesson 11
MA26200 (Gabriel Sosa), Summer ’14
Page 3 of 4
SECTION 4.3. SUBSPACES
By the end of this lesson you should be familiar with the following concepts:
1. Subspace of a Vector space.
2. Set notation for a subspace.
Additionally you should be able to:
1. Verify if a set S is a subspace of a vector space V .
2. Verify if a set S is not a subspace of a vector space V .
3. Recall that if S is a subspace of a vector space V :
→
−
(a) The zero vector 0 must be an element of S.
−
−
(b) If →
v is an element of S then −→
v must be an element of S.
Examples:
1. Is the set of all 3 × 3 skew-symmetric matrices a subspace of M3 (R)?
2. Is the set of all singular 2 × 2 matrices a subspace of M2 (R)?
3. Is S = {(ab, a, b) : a, b ∈ R} a subspace of R3 ?
4. Is S = {p(x) ∈ P4 : p(5) = 0} a subspace of P4 ?
5. Is the set of solutions of x2 y 00 + xy 0 − 4y = 0 a subspace of C 2 ([1, ∞])?
6. What about the set of solutions of x2 y 00 + xy 0 − 4x2 = 0?
Lesson 11
MA26200 (Gabriel Sosa), Summer ’14
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