Linear Independence: Combining Basis A & B to Create C

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SUMMARY

The discussion centers on the concept of linear independence in vector spaces, specifically regarding the combination of bases A and B to form a new basis C in a vector space V of dimension N. It is established that if linear independent columns from bases A and B are combined, they can indeed create a new basis C, provided that all vectors from A and B can be expressed as linear combinations of C. The key method for verifying this involves examining the null space of C and comparing it to that of A, particularly in the context of R^n.

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  • Understanding of vector spaces and their dimensions
  • Knowledge of linear independence and basis concepts
  • Familiarity with null spaces in linear algebra
  • Proficiency in R^n and its properties
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  • Study the properties of linear independence in vector spaces
  • Learn about null spaces and their significance in linear algebra
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Students and educators in linear algebra, mathematicians focusing on vector spaces, and anyone involved in theoretical mathematics or applied fields requiring an understanding of basis and linear combinations.

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Homework Statement



Suppose B and A are a basis in vector space V, dimension N, now if I can any linear independent columns of basis B and A can I combine them to make a new basis C provided that all the all, even the unused vectors of bases A & B linear combinations of C.

Homework Equations





The Attempt at a Solution


I guess the key thing to do here would be to look at the null space of C and see how it compares to A, if this is a problem in R^n then if the null spaces where the same then so would be the basis. So I guess I am just looking for a general method of figuring out if the span of two column spaces are the same, is the question posed true or false?
 
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lubuntu said:

Homework Statement



Suppose B and A are a basis in vector space V, dimension N, now if I can any linear independent columns of basis B and A can I combine them to make a new basis C provided that all the all, even the unused vectors of bases A & B linear combinations of C.
Huh? There seem to be some words missing. Please provide the exact problem description.
lubuntu said:

The Attempt at a Solution


I guess the key thing to do here would be to look at the null space of C and see how it compares to A, if this is a problem in R^n then if the null spaces where the same then so would be the basis. So I guess I am just looking for a general method of figuring out if the span of two column spaces are the same, is the question posed true or false?
 

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