Vector Subspace

Monday, July 30, 2018

9:13 AM

For a subset 𝑆 of 𝑉, we borrow the set of scalars 𝔽, additition and scalar multiplication properties from V.

If ﷐0﷯ of V is in S
𝑢+𝑣∈𝑆 for all 𝑢,𝑣∈𝑆
𝜆𝑣∈𝑆 for all 𝑣∈𝑆
Most of the axioms for a vector space are already satisfied, however we need to prove:

i.e. Show that 𝑠={𝑥∈﷐ℝ﷮3﷯:﷐𝑥﷮1﷯−2﷐𝑥﷮2﷯+3﷐𝑥﷮3﷯=1} is not a subspace of ﷐ℝ﷮3﷯

The zero vector is not in S because 0≠1
∴ S is not a subspace of ﷐ℝ﷮3﷯

A subset S of a vector space V is called a subspace of V is itself a vector space over the same field of scalars as V and under the same rules for addition and multiplication by scalars. In addition, if S is a proper subset of V (ie and S neq V), S Is called a proper subspace of V

Zero Vector
Closure under Addition
Closure under Scalar Multiplication

To prove that a subset is not a subspace, find any real example that does not meet any of the three criteria

To prove that a subset IS in a subspace, prove that the subset meets all three conditions

Subspace

A subspace S of a vector space V is a subspace if and only if

1) zero vector of V is in S
u = v in S for all u, v in s (s closed under addition)
lambda u in S for all v in s and lambda in F (s closed under scalar multiplication)

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