Simply speaking, a subset of a vector space is a subspace if is a vector space itself when inheriting the addition and scalar multiplication defined in . Definition: is a subspace of if is a non-empty subset of that satisfies the following conditions:

1. For any and in , is in .
2. For any in and real number , is in . (Note: In particular, we can set and it implies that the zero vector is in .)

Example 1: , the set containing only the zero vector, is a subspace of any vector space. It is called the zero subspace. Example 2: For any non-negative integer , , the vector space of all polynomial of degree with real coefficients, is a subspace of , the vector space of all polynomials with real coefficients. Example 3: Any plane in containing the origin is a subspace of . For example, is a subspace of because

• Zero vector is obviously in so is non-empty.
• For any , and . Then and . Therefore, is also in .
• For any real number and in i.e. . . Hence , which means is also in .

Example 4: Let be the vector space of all real number sequences and . Then is a subspace of because

• The sequence of all zeros is in so is non-empty.
• If are in , then for all even . Therefore, for all even and is also in .
• For any real number and any in , for all even . Therefore, for all even and is also in .

Example 5: Let be the vector space of all n x n matrices. Let . Then is a subspace of because

• Zero matrix is in so is non-empty.
• For any two n x n matrices in , they are upper triangular. Hence is obviously upper triangular and thus in .
• For any real number and any n x n matrix in , is upper triangular. Therefore, is also upper triangular and thus in .