Nope said:can you show me?, because I don't get the closed under addition part at all..
Nope said:yes, (2,0)
Nope said:oh ,
(1+1, 1-1)
(2,0) not in (x,y)
am i correct?
or can you explain to me in word?
Nope said:not in the set, but why is x^2+y^2 equal to one?
Nope said:is there another way to prove it? like using x1 or y2
A subspace is a subset of a vector space that satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.
Closed under addition means that when two vectors in a subspace are added together, the result must also be in the subspace. This is important because it ensures that the subspace remains a valid vector space.
Subspaces are not always closed under addition because the addition of two vectors may result in a vector that is not within the subspace. For example, if a subspace is limited to a specific dimension, adding two vectors from different dimensions may result in a vector that is not in the subspace.
To prove that a subspace is not closed under addition, we can provide a counterexample. This means finding two vectors in the subspace that, when added together, result in a vector that is not in the subspace.
No, a subspace must be closed under both addition and scalar multiplication in order to be a valid vector space. If a subspace is not closed under scalar multiplication, it may not satisfy the properties of a vector space, such as closure under addition.