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===SUBSPACE===
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To be a subspace of vectors the following must be true:
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1. One set must be a '''subset''' of another set
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2. The set must be closed under '''scalar multiplication'''
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3. The set must be closed under '''vector addition'''
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== Proving one set is a subset of another set ==
 
== Proving one set is a subset of another set ==
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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,
 
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,
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'''Basic Outline of the Proof that A is a subset of B:'''
 
'''Basic Outline of the Proof that A is a subset of B:'''
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'''·''' Suppose x ∈ A
 
'''·''' Suppose x ∈ A
  
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1. Say what it means for x to be in A;
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1. Say what it means for x to be in A
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2. Mathematical details;
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3. Conclude that x satisfies what it means to be in B;
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2. Mathematical details
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3. Conclude that x satisfies what it means to be in B
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'''·''' Conclude x∈B
 
'''·''' Conclude x∈B
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'''Example'''
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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.
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'''·''' Suppose x ∈ A:
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1. What it means for x to be in A: x = 6k for any scalar k
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2. x = 2 × (3k)
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  3k = C
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3. What it means for x to be in B: x = 2C
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'''·''' Conclude x∈B
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==Closed Under Scalar Multiplication==
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A set of vectors is closed under scalar multiplication if for every '''v'''∈V and every c∈\mathbb{R} we have c'''v'''∈V
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'''Basic Outline of the Proof V is Closed Under Scalar Multiplication:'''
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'''·''' Suppose '''v'''∈V and c∈\mathbb{R}
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1. Say what it means for '''v''' to be in V
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2. Mathematical details
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3. Conclude that c'''v''' satisfies what it means to be in V
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'''·''' Conclude c'''v'''∈V
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==Closed Under Vector Addition==
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A set of vectors is closed under vector  addition if for every '''v''' and '''w''' ∈ V we have '''v''' + '''w''' ∈ V
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'''Basic Outline of the Proof V is Closed Under Vector Addition:'''
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'''·''' Suppose '''v''' and '''w''' ∈ V
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1. Say what it means for '''v''' and '''w''' to be in V
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2. Mathematical details
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3. Conclude that '''v'''+ '''w''' satisfies what it means to be in V
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'''·''' Conclude '''v''' + '''w''' ∈ V
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'''Example'''
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Let V be the set of points above the graph of the absolute value function
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'''·''' Suppose '''v''' and '''w''' ∈ V
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1. What it means for '''v''' and '''w''' to be in V :
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'''v''' = (v1, v2) and v2 > |v1|
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'''w''' = (w1, w2) and w2 > |w1|
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2. '''z''' = (z1,z2) = '''v''' + '''w''' = (v1+w1, v2+w2)
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So, |z1| = |v1 + w1| &le; |v1| + |w1| < v2 + w2 = z2
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3. What it means for '''z''' to be in V: |z1| < z2
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'''&middot;''' Conclude '''z''' = '''v''' + '''w''' &isin; V
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----
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''Explanation of how to determine a subspace. Information referenced from Wabash College MA 223, Spring 2011''

Latest revision as of 07:48, 25 November 2012

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 above the graph of the absolute value function


· Suppose v and w ∈ V

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

v = (v1, v2) and v2 > |v1|

w = (w1, w2) and w2 > |w1|

2. z = (z1,z2) = v + w = (v1+w1, v2+w2)

So, |z1| = |v1 + w1| ≤ |v1| + |w1| < v2 + w2 = z2

3. What it means for z to be in V: |z1| < z2


· Conclude z = v + w ∈ V



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

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