Solutions to Span of Functions Problems

[gummz]

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?

 

[FactChecker]

Does that still mean that h is in the span of f and g, even though a and b are not constants?

No

 

[Mark44]

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.

 

[Bacle2]

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?

 

[blue_raver22]

Yes, it is still possible for h to be in the span of f and g even if a and b are not constants. The key concept here is that the span of a set of vectors (in this case, functions) is defined as the set of all possible linear combinations of those vectors. So as long as h can be expressed as a linear combination of f and g, it is in the span of f and g.

In this case, even though a and b are not constants, they are still coefficients that can be multiplied by f and g to create h. This means that h can still be written as a linear combination of f and g, and therefore, it is in the span of f and g.

It is important to note that the values of a and b may change the specific form of h, but as long as it can be expressed as a linear combination of f and g, it is considered to be in their span. This is a fundamental concept in linear algebra and is crucial in understanding the relationship between functions and their spans.