As we knew, a subset of a vector space is called a subspace if it is nonempty and closed under addition and scalar multiplication.
- The vector spaces are the usual Euclidean spaces of analytic geometry. There are three types of subspaces of : a line through the origin, and itself. There are four types of subspaces of : , a line through the origin, and itself.
- Let be the set of all polynomials in the single variable , with coefficients from and be the subset of consisting of all polynomials of degree or less. We have a result: is a subspace of .
- When , the set of all polynomials of degree exactly is not a subspace of .
- Let be the set of all functions ( is field or ). is a vector space. Why?
- have many subspaces, for example, we consider the case of and . The set of all continuous functions is a subspace of . The set of all differentiable functions is a subspace and also of . Prove that?
D. Lay, Linear algebra and its applications, Addison-Wesley, 2012.
Nguyen Huu Viet Hung, Linear algebra (Vietnamese), VNU Publisher .