The discussion revolves around the concept of vector span in the context of linear algebra, specifically whether two independent vectors can span R². The original poster questions how two vectors that are not scalar multiples of each other can span a plane, and whether this implies they span R².
There is an ongoing exploration of the definitions and assumptions related to vector span. Some participants suggest that the vectors must be in R² to span R², while others question the dimensionality based on the vectors' entries. The discussion reflects a mix of interpretations regarding the relationship between vector entries and the concept of span.
Participants note that the vectors' dimensionality and the definition of R² may vary based on context, leading to differing opinions on whether the span can be considered R² or a plane within a higher-dimensional space.
flyingpig said:I thought Span is the linear combination of all vectors in that set and hence it must be R^2?
Assuming v1 and v2 are in R^2, if v1 = c v2 for some c in R, then no, span{v1, v2} is not equal to R^2.Theorem. said:your set consists of two vectors, v1 and v2. if v1 and v2 are both elements of R^2, then they will span all of R^2.
Unit said:Assuming v1 and v2 are in R^2, if v1 = c v2 for some c in R, then no, span{v1, v2} is not equal to R^2.
Theorem. said:your set consists of two vectors, v1 and v2. if v1 and v2 are both elements of R^2, then they will span all of R^2. nowhere have you stated that v1 and v2 are both elements of R^2, they could be vectors in R^3.
Unit said:My apologies, Theorem; I was hasty. You are right.
Theorem. said:No worries : ) hopefully the OP understands what i meant
Two independent vectors will always span a plane. Whether or not that plane is R2 or not depends upon the vectors and what you mean by R2. The two vectors <1, 0, 1> and <0, 1, 0> will have span a<1, 0, 1>+ b<0, 1, 0>= <a, b, a> which can be expressed as <x, y, z> with z= x, or the plane z= x.flyingpig said:Does it matter how many entries I have in my vectors? I can have three entries in my vectors, but with two vectors, they will always span R2.
Is there even any relations to having free variables (parameters) with the concept of Span?
The definition of "span" of a set of vectors is the set of all linear combinations of the vectors. In particular, the span of the two vectors {u, v} is all vectors of the form au+ bv for any scalars a and b. Both a and b are, I think, what you are calling "parameters". Where did you get the idea that there could be "at most one parameter"? The span of a set of n vectors will involve n parameters and, if the vectors are independent, the span will be an n-dimensional subspace.So the question is, how? There can be at most one parameter right? One parameter means the solution is a line, we need two to make a plane?