The discussion revolves around determining the span of the set of vectors {(1; 1; 0), (0; 0; 2)} in a linear algebra context. Participants are exploring how to express the span and its implications in terms of linear combinations of the given vectors.
The discussion is active, with participants providing insights into the nature of the span and questioning the notation used in the answer. Some clarification has been offered regarding the scalar relationship between the vectors, but no consensus has been reached on the final expression of the span.
Participants express a desire to understand the reasoning behind the notation used in the answer provided on the homework sheet. There is an acknowledgment of the potential for multiple valid representations of the span.
The span is all linear combinations of <1, 1, 0> and <0, 0, 2>, which is the same span as all linear combinations of <1, 1, 0> and <0, 0, 1>.negation said:Homework Statement
What is the span of {(1; 1; 0),(0; 0; 2)}?
The Attempt at a Solution
So the span is λ1(1; 1; 0) + λ2(0; 0; 2)
negation said:But how should I express my answer?
The given answer on the sheet is as below:
The span is {( λ,λ,β )|λ,βεR}
Mark44 said:The span is all linear combinations of <1, 1, 0> and <0, 0, 2>, which is the same span as all linear combinations of <1, 1, 0> and <0, 0, 1>.
Using the same letters as in the answer, the span is all vectors of the form λ<1, 1, 0> + β<0, 0, 1> = <λ, λ, 0> + <0, 0, β>, or more simply, <λ, λ, β>, with λ and β arbitrary reals.
Yes, <0, 0, 2> is a scalar multiple of <0, 0, 1>. You could just as well have picked <0, 0, 17>.negation said:In red, where did <0,0,1> came from? 2<0,0,1> = <0,0,2>?
negation said:Edit: Alright, let's not complicate things since my intention was only to know why the answer was expressed in the way it was.
Thanks