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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 | 1. Say what it means for x to be in A | ||
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'''Example''' | '''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. | 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: | ||
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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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| + | ==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 | ||
| + | |||
| + | 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 | ||
Revision as of 07:21, 25 November 2012
Contents
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
