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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 == | ||
Revision as of 07:08, 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
