Linear Algebra Done Right- Chapter 1C

Subspaces

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

conditions for a subspace

A subset 𝑈 of 𝑉 is a subspace of 𝑉 if and only if 𝑈 satisfies the following three conditions.

additive identity

0 ∈ 𝑈.

closed under addition

𝑢, 𝑤 ∈ 𝑈 implies 𝑢 + 𝑤 ∈ 𝑈.

closed under scalar multiplication

𝑎 ∈ 𝐅 and 𝑢 ∈ 𝑈 implies 𝑎𝑢 ∈ 𝑈.

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 condi- tion 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 multiplica- tion makes sense on 𝑈.

If 𝑢 ∈ 𝑈, then −𝑢 [which equals (−1)𝑢 by 1.32] 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.

sum of subspaces

Suppose 𝑉1, ... , 𝑉𝑚 are subspaces of 𝑉. The sum of 𝑉1, ... , 𝑉𝑚, denoted by 𝑉1 +⋯+𝑉𝑚, is the set of all possible sums of elements of 𝑉1,...,𝑉𝑚. More precisely, 𝑉1 +⋯+𝑉𝑚 ={𝑣1 +⋯+𝑣𝑚 ∶𝑣1 ∈𝑉1,...,𝑣𝑚 ∈𝑉𝑚}.

sum of subspaces is the smallest containing subspace

Suppose 𝑉1, ... , 𝑉𝑚 are subspaces of 𝑉. Then 𝑉1 + ⋯ + 𝑉𝑚 is the smallest subspace of 𝑉 containing 𝑉1, ... , 𝑉𝑚.

Proof The reader can verify that 𝑉1 + ⋯ + 𝑉𝑚 contains the additive identity 0 and is closed under addition and scalar multiplication. Thus 1.34 implies that 𝑉1 +⋯+𝑉𝑚 isasubspaceof𝑉.

The subspaces 𝑉1, ... , 𝑉𝑚 are all con- tained in 𝑉1 + ⋯ + 𝑉𝑚 (to see this, con- sider sums 𝑣1 + ⋯ + 𝑣𝑚 where all except one of the 𝑣𝑘’s are 0). Conversely, every subspace of 𝑉 containing 𝑉1, ... , 𝑉𝑚 con- tains 𝑉1 + ⋯ + 𝑉𝑚 (because subspaces must contain all finite sums of their ele- ments). Thus 𝑉1 + ⋯ + 𝑉𝑚 is the smallest subspace of 𝑉 containing 𝑉1, ... , 𝑉𝑚.

direct sum, ⊕

Suppose 𝑉1, ... , 𝑉𝑚 are subspaces of 𝑉.

• The sum 𝑉1 +⋯+𝑉𝑚 is called a direct sum if each element of 𝑉1 +⋯+𝑉𝑚 can be written in only one way as a sum 𝑣1 +⋯+𝑣𝑚, where each 𝑣𝑘 ∈𝑉𝑘.

• If 𝑉1 +⋯+𝑉𝑚 is a direct sum, then 𝑉1 ⊕⋯⊕𝑉𝑚 denotes 𝑉1 +⋯+𝑉𝑚, with the ⊕ notation serving as an indication that this is a direct sum.

example: a sum that is not a direct sum

condition for a direct sum

Suppose 𝑉1,...,𝑉𝑚 are subspaces of 𝑉. Then 𝑉1 +⋯+𝑉𝑚 is a direct sum if and only if the only way to write 0 as a sum 𝑣1+⋯+𝑣𝑚, where each 𝑣𝑘 ∈𝑉𝑘, is by taking each 𝑣𝑘 equal to 0.

direct sum of two subspaces

Suppose 𝑈 and 𝑊 are subspaces of 𝑉. Then

𝑈+𝑊 is a direct sum ⟺ 𝑈∩𝑊={0}.