Proving the Existence of Direct Sums in Linear Algebra

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SUMMARY

The discussion centers on the definition of direct sums in linear algebra as presented in 'Linear Algebra Done Right' by Sheldon Axler. A vector space V is defined as the direct sum of subspaces U_1, ..., U_m if every element in V can be expressed uniquely as a sum of elements from these subspaces. The user seeks to prove that for any vector u in subspace U, there exists a vector v in V and a vector w in W such that v = u + w, thereby facilitating the solution to a related problem in the text.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with the concept of direct sums in linear algebra
  • Knowledge of unique representation of vectors in linear combinations
  • Basic proficiency in reading mathematical proofs and definitions
NEXT STEPS
  • Study the properties of direct sums in linear algebra
  • Learn about the unique representation of vectors in subspaces
  • Explore the implications of direct sums on vector space dimensions
  • Investigate examples of direct sums in practical applications
USEFUL FOR

Students of linear algebra, educators teaching vector space concepts, and mathematicians interested in the foundational aspects of vector spaces and their properties.

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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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Take [itex]w=0[/itex] and [itex]v=u[/itex].
 
LOL, OF COURSE!

Thanks!
 

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