Proving Linear Independence: Vectors in R^5 and Their Span

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SUMMARY

The discussion centers on proving that at least one of the linearly independent vectors v1, v2, and v3 in R^5 is not contained within the span of the vectors a1, a2, and a3, defined as a1 = v1 + v2 - 2v3, a2 = 3v1 + v2 + 4v3, and a3 = v1 + 2v2 - 7v3. The transformation matrix T = {{1,1,-2},{3,1,4},{1,2,-7}} is suggested for use in row reduction to analyze the relationship between the vectors. The goal is to establish the independence of the original vectors from the span of the newly defined vectors.

PREREQUISITES
  • Understanding of linear independence in vector spaces
  • Familiarity with R^5 vector notation
  • Knowledge of matrix row echelon form and transformations
  • Experience with vector spans and their properties
NEXT STEPS
  • Learn how to perform row reduction on matrices to determine linear independence
  • Study the concept of vector spans in R^n spaces
  • Explore the implications of linear transformations on vector sets
  • Investigate the properties of bases in higher-dimensional vector spaces
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, vector spaces, and transformations, will benefit from this discussion.

NeonVomitt
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Suppose that v1,v2,v3 are linearly independent vectors in R^5 and consider the vectors a1,a2,a3 defined by a1=v1+v2-2v3, a2=3v1+v2+4va, and a3=v1+2v2-7v3. Show that at least one of the vectors v1,v2,v3 is not in the span of the vectors a1,a2,a3.

I am kind of confused. Should I somehow reduce row echelon it? But how would I even set that up given this type of format?

Thank you!
 
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a=T.v
where
T={{1,1,-2},{3,1,4},{1,2,-7}}
 

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