Suppose that x and y are vectors and M is a subspace in a vector space V;
let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
Prove that if y is in K but not in M, then x is in K.
So, K is the set of all linear combinations of elements in M and the vector y.
K is also the set of all linear combination of elements in M and the vector x.
This definition of K makes me think that x is always in K.
What am I missing here?