Relation between Basis of a Vector Space and a Subspace

Question

$V$ is a vector space, and $H\subset V$ is a subspace of $V$. If $\beta$ is a basis for V, can we guarantee that $\beta \cap H$ is a basis for H?

Answer 1

No, you cannot. For example, take $V = \Bbb R^2$. The space $H = \{(t,t): t \in \Bbb R\}$ is a subspace, and $\beta = \{(1,0),(0,1)\}$ is a basis. However, $\beta \cap H = \emptyset$ is not a basis for $H$.

We can say, however, that any basis for $H$ can be extended to a basis for $V$. Also, $\beta \cap H$ is necessarily a linearly independent set, and will generally form the basis for a subspace of $H$.

Answer 2

$\mathbb R^2$ is a vector space. $(1, 1)$ and $(1, -1)$ form a basis. H = $\{ (x, 0) \mid x \in \mathbb R \}$ is a subspace (the "x-axis"). But $H \cap \beta = \emptyset$ is not a basis for $H$.