(True or false.) Let V be an n-dimensional vector space, and let S be a k-dimensional subspace of V, where k

Question

(True or false.) Let V be an n-dimensional vector space, and let S be a k-dimensional subspace of V, where k < n. If {⃗v1,...,⃗vk} is a basis for S and {⃗vk+1,...,⃗vn} is a set of n − k linearly independent vectors not in S, then {v1,...,⃗vk,⃗vk+1,...,⃗vn} is a basis for V.

anonymous · Answer

It is indeed false.

Example:
$V = \mathbb{R}^3$

$v_1 = (1, 0, 0)$
$S = \text{span}\ \{v_1\}$

$v_2 = (1, 1, 1)$
$v_3 = (0, 1, 1)$

Notice that $\{v_2, v_3\}$ is linearly independent and $v_2, v_3 \not\in S$. Yet $\{v_1, v_2, v_3\}$ is not linearly independent since $v_1 = v_2 - v_3$. Therefore $\{v_1, v_2, v_3\}$ is not a basis for $\mathbb{R}^3$.