Linear Independence and Spanning

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SUMMARY

The discussion confirms that if the set of vectors {W1, W2, W3, W4} is linearly independent in R4, then they span R4. Consequently, any vector, including [2, tan(h), 7, 4sec(k)], cannot exist outside the span of these vectors. Therefore, it is impossible to find values for h and k that would place this vector outside the span of {W1, W2, W3, W4}. The conclusion is that the original assertion about the impossibility of finding such values is correct.

PREREQUISITES
  • Understanding of linear independence in vector spaces
  • Knowledge of vector spanning in R4
  • Familiarity with trigonometric functions such as tangent and secant
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the properties of linear independence and spanning sets in vector spaces
  • Explore the implications of the Rank-Nullity Theorem in linear algebra
  • Learn about the geometric interpretation of vectors in R4
  • Investigate the relationship between linear transformations and vector spans
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching concepts of vector spaces and linear independence.

Jack Nagel
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Say that {W1, W2, W3, W4} is linearly independent in R4.

Now say I have this vector
[ 2
tan(h)
7
4sec(k)
]

and I want to find values of h and k such that it is not in the span of (W1...W4).

If I understand this correctly, it means it is impossible to find those values since they do not exist. If a set of 4 vectors in R4 are linearly independent, then they also span R4.

Am I right or way off on this?
 
Physics news on Phys.org
That sounds right.
 

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