Solutions to Span of Functions Problems

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SUMMARY

For a function h to be in the span of two other functions f and g, it must be expressed as a linear combination of these functions, specifically in the form h = af + bg, where a and b are constants. If a and b are not constants, then h is not in the span of f and g. This definition aligns with the concept of vectors in linear algebra, where a vector must also be a linear combination of other vectors to belong to their span. Further clarification on the vector space in question would enhance understanding of the relationship between the functions involved.

PREREQUISITES
  • Understanding of linear combinations in vector spaces
  • Familiarity with the concept of spans in linear algebra
  • Knowledge of function representation and manipulation
  • Basic principles of vector spaces and bases
NEXT STEPS
  • Study the definition and properties of vector spans in linear algebra
  • Learn about linear combinations and their applications in function analysis
  • Explore the relationship between functions and vector spaces in mathematical contexts
  • Investigate specific examples of function spans in various vector spaces
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Mathematicians, students of linear algebra, and anyone interested in the properties of functions and their relationships in vector spaces.

gummz
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So let's say I have a function that I want to find out if is in the span of two other functions, for example, a*f + b*g = h, where f, g, and h are functions, and a and b are constants. Let's say I find a solution where a and b are not constants. Does that still mean that h is in the span of f and g, even though a and b are not constants?
 
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gummz said:
Does that still mean that h is in the span of f and g, even though a and b are not constants?
No
 
For a function h to be in the span of two other functions f and g, h must be a linear combination of f and g. IOW, h = af + bg, where a and b are constants. It's almost exactly the same definition for a vector to be in the span of two other vectors.
 
It would be nice if the OP could be more specific about the vector space s/he is working in; gummz, can
you tell us more about what space you are working in? are g,h part of a basis for the space?
 

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