How can I identify the vector spaces whether it has span & basis or not?
Following which vector Spaces have span & basis?
hi
I am not going to just give you straight answers. What have you learnt about spanning sets, and bases? When does a set of vectors form a basis for a vector space?
Hint: Vectors must be linearly independent.
Hint: If a set contains just one vector, it is linearly independent if and only if that vector is not the zero vector.
Hi
1. The vectors are not multiples of each other so they are lin.independent.
They form a basis for Span{v1,v2}. They span a plane in R^3.
2. The vectors are multiples of each other, so they are not lin.independent, hence they do not form a basis. They span a line through the origin in R^2.
(Only one of the vectors is needed, since Span{v1,v2}=Span{v1},because
v2=k*v1, for some k).
3. Same as number 2.
4. You see that vector one and two in your set are lin.independent, and they span R^2 ->They form a basis for R^2. This also means that there must exist scalars c1,c2 such that v3=c1*v1+c2*v2, which means that the set
{v1,v2,v3} cannot possibly be linearly independent, which also gives that
they do not form a basis. The vectors span R^2, but do not form basis.
5. Same as 2. They are not lin.indept. -> do not form basis.
They dont span R^2.
6. This set consisting of only one vector is linearly independent. (Why?).
So this vector span a subspace to R^2.