Some examples of subspaces – 1

As we knew, a subset {W} of a vector space {V} is called a subspace if it is nonempty and closed under addition and scalar multiplication.

  • The vector spaces {\mathbb{R}, \mathbb{R}^2, \mathbb{R}^3} are the usual Euclidean spaces of analytic geometry. There are three types of subspaces of {\mathbb{R}^2}: {\{\theta\},} a line through the origin, and {\mathbb{R}^2} itself. There are four types of subspaces of {\mathbb{R}^3}: {\{\theta\}}, a line through the origin, and {\mathbb{R}^3} itself.
  • Let {P[x]} be the set of all polynomials in the single variable {x}, with coefficients from {\mathbb{R}} and {P_n[x]} be the subset of {P[x]} consisting of all polynomials of degree {n} or less. We have a result: {P_n[x]} is a subspace of {P[x]}.
  • When {n > 0}, the set of all polynomials of degree exactly {n} is not a subspace of {P[x]}.
  • Let {\mathcal{F}(X, F)} be the set of all functions {f: X \rightarrow F} ({F} is field {\mathbb{R}} or {\mathbb{C}}). {\mathcal{F}(X, F)} is a vector space. Why?
  • {\mathcal{F}(X, F)} have many subspaces, for example, we consider the case of {X \subset \mathbb{R}^n} and {F = \mathbb{R}}. The set {C(X)} of all continuous functions {f: X \rightarrow \mathbb{R}} is a subspace of {\mathcal{F}(X, \mathbb{R})}. The set {\mathcal{D}(X)} of all differentiable functions {f: X \rightarrow \mathbb{R}} is a subspace {C(X)} and also of {\mathcal{F}(X, \mathbb{R})}. Prove that?

References

D. Lay, Linear algebra and its applications, Addison-Wesley, 2012.

Nguyen Huu Viet Hung, Linear algebra (Vietnamese), VNU Publisher .

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