Difference between revisions of "Subspace MA265F12Alvarado" - Rhea

Revision as of 11:23, 3 December 2012

VECTOR SPACE

A vector space is a set of vectors that defines addition V x V --> V and scalar multiplcation cV --> V that satisfy the following properties:

1. Communative Property: 'u + v = v + u'

2. Associative Property:

a. Of addition:'(u + v) + w = u = (v + w)'

b. Of multiplication: '(ab)v = a(bv)'

3. Zero Property: There exist some 0x∈V such that '0 + v = v'

4. Inverse Property: For every v∈V there is some w∈V such that 'v+w=0'

5. Identity Property: '1v=v'

6. Distributive Property: 'a(u + v) = au + av

SUBSPACE

To be a subspace of vectors the following must be true:

1. One set must be a subset of another set

2. The set must be closed under scalar multiplication

3. The set must be closed under vector addition

Proving one set is a subset of another set

Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is,

x∈A implies x∈B

Basic Outline of the Proof that A is a subset of B:

· Suppose x ∈ A

1. Say what it means for x to be in A

2. Mathematical details

3. Conclude that x satisfies what it means to be in B

· Conclude x∈B

Example

Let A be the set of scalars divisible by 6 and let B be the even numbers. Prove that A is a subset of B.

· Suppose x ∈ A:

1. What it means for x to be in A: x = 6k for any scalar k

2. x = 2 × (3k)

  3k = C

3. What it means for x to be in B: x = 2C

· Conclude x∈B

Closed Under Scalar Multiplication

A set of vectors is closed under scalar multiplication if for every v∈V and every c∈\mathbb{R} we have cv∈V

Basic Outline of the Proof V is Closed Under Scalar Multiplication:

· Suppose v∈V and c∈\mathbb{R}

1. Say what it means for v to be in V

2. Mathematical details

3. Conclude that cv satisfies what it means to be in V

· Conclude cv∈V

Closed Under Vector Addition

A set of vectors is closed under vector addition if for every v and w ∈ V we have v + w ∈ V

Basic Outline of the Proof V is Closed Under Vector Addition:

· Suppose v and w ∈ V

1. Say what it means for v and w to be in V

2. Mathematical details

3. Conclude that v+ w satisfies what it means to be in V

· Conclude v + w ∈ V

Example

Let V be the set of points in R^2 such that x=y

· Suppose v and w ∈ V

1. What it means for v and w to be in V :

v = (v1, v2) and v1 = v2

w = (w1, w2) and w1 = w2

2. z = v + w = (v1+w1, v2+w2) = (v1+w1, v1+w1)

3. What it means for z to be in V: v1+w1 = v2+w2

· Conclude z = v + w ∈ V

Explanation of how to determine a subspace. Information referenced from Wabash College MA 223, Spring 2011