Linear Algebra Proof of Span and orthogonal vector space

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SUMMARY

The discussion centers on proving that if a vector v is orthogonal to all vectors w_i in a span W = span(w_1, w_2, ..., w_k) in R^n, then v belongs to the orthogonal complement of W, denoted as W⊥. The proof utilizes the definition of the dot product as the inner product in R^n, establishing that for any vector x in W, expressed as a linear combination of the w_j, the condition v·x = 0 holds true. This confirms that v is indeed an element of W⊥.

PREREQUISITES
  • Understanding of vector spaces in R^n
  • Knowledge of the concept of span in linear algebra
  • Familiarity with the dot product as an inner product
  • Basic principles of orthogonal vectors and orthogonal complements
NEXT STEPS
  • Study the properties of orthogonal complements in vector spaces
  • Explore the concept of linear independence and dependence
  • Learn about the Gram-Schmidt process for orthogonalization
  • Investigate applications of orthogonal projections in linear algebra
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching concepts related to vector spaces and orthogonality.

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Let w=span(w1, w2, ...,wk) where wi are vectors in R^n. Let dot product be inner product for R^n here. Prove that if v*wi=0 for all i-1,2,...,k then v is an element of w^upside down T (w orthogonal).
 
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Take an element x of W. You must prove that v.x=0.

Try to prove this by expressing x as a linear combination of the wj.
 

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