A subspace is a subset of a vector space that contains all the properties of the original vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.
To prove that W is a subspace, we need to show that it satisfies the three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done by using the definition of a subspace and showing that all the properties hold for the elements in W.
Closure under addition means that when we add two vectors in W, the resulting vector must also be in W. This ensures that the subspace remains within the original vector space and does not contain any elements that do not belong to it.
Closure under scalar multiplication means that when we multiply a vector in W by a scalar, the resulting vector must also be in W. This ensures that the subspace is closed under all possible scalar operations and does not contain any elements that do not belong to it.
To show that W contains the zero vector, we need to demonstrate that the zero vector is an element of W. This can be done by showing that the zero vector satisfies all the properties of the subspace, such as closure under addition and scalar multiplication.