Linear Independence of Sets in a Linear Space

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SUMMARY

The discussion centers on the concept of linear independence within a linear space V, specifically examining the sets {u, v, w} and {u, u+v, u+v+w}. It is established that if the set {u, v, w} is linearly independent, then the set {u, u+v, u+v+w} is also linearly independent. Additionally, the notation U ∩ V ≠ {OV} is questioned, with OV referring to the zero vector in the space V, indicating a need for clarification on the definitions of U and OV.

PREREQUISITES
  • Understanding of linear spaces and their properties
  • Familiarity with the concept of linear independence
  • Knowledge of vector notation and operations
  • Basic understanding of set theory in the context of linear algebra
NEXT STEPS
  • Study the properties of linear independence in vector spaces
  • Learn about the implications of linear combinations in linear algebra
  • Explore the definitions and roles of zero vectors in linear spaces
  • Investigate the intersection of subspaces in linear algebra
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts of linear independence and vector spaces.

mrs.malfoy
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Let V be a linear space and u, v, w \in V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}
 
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mrs.malfoy said:
Let V be a linear space and u, v, w \in V. Show that U \cap V is not equal to {OV}
?? What is U?
 
Can you explain your notation?
So u is an element of the linear space V, but what is U?
What is OV, is it the zero element in V?
 

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