A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. In other words, if two vectors are in the subspace, their sum and any scalar multiple of them must also be in the subspace.
To prove that a set is a subspace, you must show that it satisfies the three conditions of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that any two vectors in the set can be added together and multiplied by a scalar to still be in the set, and that the zero vector is also in the set.
A vector space is a set of vectors that satisfy certain properties, such as closure under addition and scalar multiplication, and containing a zero vector. A subspace is a subset of a vector space that also satisfies these properties. In other words, all subspaces are vector spaces, but not all vector spaces are subspaces.
Yes, a subspace can have any number of dimensions, as long as it satisfies the three conditions of a subspace. For example, a plane in three-dimensional space is a subspace with two dimensions.
To use the U + v in U property to prove that a set is a subspace, you must show that for any two vectors u and v in the set, their sum u + v is also in the set. This can be done by showing that the set is closed under addition, and that any scalar multiple of a vector in the set is also in the set.