Vector subspace as the space of solutions to matrix multiplication

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SUMMARY

The discussion centers on proving that a subset W of a vector space V = F^n is a subspace of solutions to the matrix equation AX = 0 for some matrix A. The key approach involves defining the orthogonal complement of W and utilizing the projection onto that complement to construct matrix A. This method effectively demonstrates the relationship between subspaces and solutions to linear equations in vector spaces.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Knowledge of matrix equations and their solutions
  • Familiarity with orthogonal complements in linear algebra
  • Proficiency in matrix projection techniques
NEXT STEPS
  • Study the properties of orthogonal complements in vector spaces
  • Learn about matrix projections and their applications
  • Explore the implications of the Rank-Nullity Theorem
  • Investigate the relationship between linear transformations and matrix representations
USEFUL FOR

Students and educators in linear algebra, mathematicians focusing on vector spaces, and anyone interested in the theoretical foundations of matrix equations.

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Given a subset W of a vector space V = F^n (for some field F), prove that W is the subspace of solutions of the matrix equation AX = 0 for some A.
 
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Why?

Seriously, this sounds like a homework problem so it should be in the homework section.

In any case, define the orthogonal complement of W and let A be the projection onto that complement.
 
I would just solve for A myself. :smile:
 

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