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'''·''' Suppose x ∈ A
 
'''·''' Suppose x ∈ A
 
  
 
1. Say what it means for x to be in A
 
1. Say what it means for x to be in A
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'''Example'''
 
'''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.
 
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:
 
'''·''' Suppose x ∈ A:
 
  
 
1. What it means for x to be in A: x = 6k for any scalar k
 
1. What it means for x to be in A: x = 6k for any scalar k
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'''·''' Conclude x∈B
 
'''·''' Conclude x∈B
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 +
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==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 c'''v'''∈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 c'''v''' satisfies what it means to be in V
 +
 +
 +
'''·''' Conclude c'''v'''∈V
 +
 +
 +
==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
 +
 +
 +
'''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
 +
 +
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'''·''' Conclude '''v''' + '''w''' ∈ V

Revision as of 08:21, 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

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett