2.2. Subspace

(1) , U => + U,
and
(2) U, a K => a U.

Proposition 1. If U is a subspace of V, then

(i) U

(ii) a₁ , a₂ , ..., a_n K, ₁ , ..., _n U => a_{i i} U

(iii) U is a vector space

Proof. (i): U nonempty => U
(2) => = · U
(ii): (1), (2) => a_{1 1} + a_{2 2} U, and the statement follows by induction.
(iii) U satisfies the vector space axioms 1) - 8)

Example 1: A subspace

The subspace spanned by S V , denoted by L (S), is the set of all finite linear combinations formed from the elements of S:

L (S) = {y | y = a_{i i} , _i S, a_i K, 1 n < }

Proposition 2. L (S) is a subspace.

Proposition 3. The sum

U₁ + U₂ = {₁ + ₂ | ₁ U₁ , ₂ U₂}

of two subspaces U₁ and U₂ is a subspace. So is also U₁ U₂ .

Proof. , U₁ + U₂ => = ₁ + ₂    = ₁ + ₂
=>
1) + = (₁ + ₁ ) + (₂ + ₂ ) U₁ + U₂
2) a = a₁ + a₂ U₁ + U₂
=>
U₁ + U₂ is a subspace.

Example 2: A subspace

If U₁ , U₂ are subspaces withU₁ U₂ = {}, the sum U₁ + U₂ is denoted by U₁ U₂ and it is said to be a direct sum.