Proving the Existence of Direct Sums in Linear Algebra

In summary, a direct sum in linear algebra is when a vector space can be written as the unique sum of subspaces. In order to prove that for all u in U, there exists v in V and w in W such that v = u + w, one can simply take w=0 and v=u. This can be useful in solving problems later in the book.
  • #1
omoplata
327
2
In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way,

We say that [itex]V[/itex] is the direct sum of subspaces [itex]U_1, \dotsc ,U_m[/itex] written [itex]V = U_1 \oplus \dotsc \oplus U_m[/itex], if each element of [itex]V[/itex] can be written uniquely as a sum [itex]u_1 + \dotsc + u_m[/itex], where each [itex]u_j \in U_j[/itex].

Suppose [itex]V = U \oplus W[/itex]. Is there any way I can prove that for all [itex]u \in U[/itex] there exists [itex]v \in V[/itex] and [itex]w \in W[/itex] such that [itex]v = u + w[/itex]?

If that can be done, then I can solve a problem given later in the book.
 
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  • #2
Take [itex]w=0[/itex] and [itex]v=u[/itex].
 
  • #3
LOL, OF COURSE!

Thanks!
 

1. What is a direct sum of vector spaces?

A direct sum of vector spaces is a mathematical operation that combines multiple vector spaces into a single vector space. It is denoted by the symbol ⊕ and is also known as the direct product of vector spaces. It is a way of representing the combination of all possible linear combinations of vectors from each individual space.

2. How is a direct sum different from a direct product?

A direct sum and a direct product are two different mathematical operations involving vector spaces. While a direct sum combines vector spaces to create a new vector space, a direct product combines elements from each individual vector space to create a new set. In simpler terms, a direct sum creates a new space, while a direct product creates a new set.

3. What are the properties of a direct sum?

The properties of a direct sum include closure, associativity, and commutativity. This means that the direct sum of two vector spaces will always result in a new vector space, the order in which the vector spaces are combined does not matter, and the resulting vector space will remain the same regardless of the order in which the individual spaces are combined.

4. How is a direct sum related to linear independence?

A direct sum is closely related to the concept of linear independence. If two vector spaces are linearly independent, then their direct sum will also be linearly independent. This means that the vectors in the direct sum cannot be written as linear combinations of vectors from the individual spaces. In other words, the direct sum of linearly independent vector spaces creates a new space with a greater number of linearly independent vectors.

5. Can a direct sum have more than two vector spaces?

Yes, a direct sum can have any number of vector spaces. The process of combining multiple vector spaces remains the same regardless of the number of spaces involved. The resulting vector space will contain all possible linear combinations of vectors from each individual space. This makes direct sums a useful tool in mathematics for creating new vector spaces with unique properties.

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