- 71

- 10

## Main Question or Discussion Point

Let ##\mathbb{V}## be a vector space and ##\mathbb{W}## be a subset of ##\mathbb{V}##, with the same operations.

Claim:

If ##\mathbb{W}## is non-empty, closed under addition and scalar multiplication, then ##\mathbb{W}## is a subspace of ##\mathbb{V}##.

A set is a vector space if it satisfies 10 properties:

The properties 1 and 2 are given. 3,4,7,8,9,10 are easily verified as ##\mathbb{W}## and ##\mathbb{V}## share same operations. For example 3:

Let ##\vec u , \vec v \epsilon \mathbb{W}##

$$ \vec u + \vec v (in~ \mathbb{W}) \\

= \vec u + \vec v (in~ \mathbb{V}) \text{(same operation)} \\

= \vec v + \vec u (in~ \mathbb{V}) (\mathbb{V} \text{is a vector space.)} \\

= \vec v + \vec u (in~ \mathbb{W}) ~\text{(same operation)} \\

$$

The problem is 5 and 6.

The book I'm using ( S. Andrilli and D. Hecker) says:-

Let ##\vec u~ \epsilon ~\mathbb{W}##

$$ \Rightarrow~~ 0\vec u ~ \epsilon ~\mathbb{W} ~\text{(closed under scalar multiplication.)}\\

\Rightarrow ~~ 0\vec u ~ \epsilon ~ \mathbb{V} ~\text{(property of subset)}\\

\Rightarrow ~~ \vec 0 ~\epsilon ~ \mathbb{V} ~\text{(property of vector space. Which we proved earlier.)}\\

\Rightarrow ~~ \vec 0 ~ \epsilon ~ \mathbb{W}~ \text{(as they share the same operations)}$$

what??

How can same operations imply that if ##\vec 0## is in ##\mathbb{V}## it also must be in ##\mathbb{W}## ??

Claim:

If ##\mathbb{W}## is non-empty, closed under addition and scalar multiplication, then ##\mathbb{W}## is a subspace of ##\mathbb{V}##.

A set is a vector space if it satisfies 10 properties:

- Closure under addition
- Closure under scalar multiplication
- Commutativity under addition
- Associativity under addition
- Existence of additive identity
- Existence of additive inverse
- Distributivity for scalar multiplication over addition in scalars
- Distributivity for scalar multiplication over addition in vectors
- Associativity under scalar multiplication
- Identity for scalar multiplication

The properties 1 and 2 are given. 3,4,7,8,9,10 are easily verified as ##\mathbb{W}## and ##\mathbb{V}## share same operations. For example 3:

Let ##\vec u , \vec v \epsilon \mathbb{W}##

$$ \vec u + \vec v (in~ \mathbb{W}) \\

= \vec u + \vec v (in~ \mathbb{V}) \text{(same operation)} \\

= \vec v + \vec u (in~ \mathbb{V}) (\mathbb{V} \text{is a vector space.)} \\

= \vec v + \vec u (in~ \mathbb{W}) ~\text{(same operation)} \\

$$

The problem is 5 and 6.

The book I'm using ( S. Andrilli and D. Hecker) says:-

Let ##\vec u~ \epsilon ~\mathbb{W}##

$$ \Rightarrow~~ 0\vec u ~ \epsilon ~\mathbb{W} ~\text{(closed under scalar multiplication.)}\\

\Rightarrow ~~ 0\vec u ~ \epsilon ~ \mathbb{V} ~\text{(property of subset)}\\

\Rightarrow ~~ \vec 0 ~\epsilon ~ \mathbb{V} ~\text{(property of vector space. Which we proved earlier.)}\\

\Rightarrow ~~ \vec 0 ~ \epsilon ~ \mathbb{W}~ \text{(as they share the same operations)}$$

what??

How can same operations imply that if ##\vec 0## is in ##\mathbb{V}## it also must be in ##\mathbb{W}## ??