Lecture 2

Chapter I Vector Spaces

Subspaces 1C

Definition 1.33

A subset U of V is called subspace of V is U is also a vector space with the same additive identity, addition and scalar multiplication as on V.

Theorem 1.34

Condition for a subspace.

Additive identity: 0\in U
Closure under addition: \forall u,w\in U,u+w\in V
Closure under scalar multiplication: a\in \mathbb{F} and u\in V, a\cdot u\in V

Proof If U is a subspace of V, then U satisfies the three conditions above by the definition of vector space.

Conversely, suppose U satisfies the three conditions above. The first condition ensures that the additive identity of V is in U.

The second condition ensures that addition makes sense on U. The third condition ensures that scalar multiplication makes sense on U.

If u\in U, then -u is also in U by the third condition above. Hence every element of U has an additive inverse in U. The other parts of the definition of a vector space, such as associativity and commutativity, are automatically satisfied for U because they hold on the larger space V. Thus U is a vector space and hence is a subspace of V.

Definition 1.36

Sum of subspaces

Suppose V_1,...,V_m are subspace of V. The sum of V_1,...,V_m, denoted by V_1+...+V_m is the set of all possible sum of elements of V_1,...,V_m.


V_1+...+V_m=\{v_1+...+v_m:v_1\in V_1, ..., v_m\in V_m\}

Example

a sum of subspaces of \mathbb{F}^3

Suppose U is the set of all elements of \mathbb{F}^3 whose second and third coordinates equal 0, and 𝑊 is the set of all elements of \mathbb{F}^3 whose first and third coordinates equal 0:


U = \{(x,0,0) \in \mathbb{F}^3 : x\in \mathbb{F}\} \textup{ and } W = \{(0,y,0) \in \mathbb{F}^3 :y\in  \mathbb{F}\}.

Then


U+W= \{(x,y,0)  \in \mathbb{F}^3 : x,y \in \mathbb{F}\}

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