Let $\{v_1, v_2, ... , v_k\}$ be a linearly independent set of vectors in a vector space V . Show that the set $\{v_1, v_2, ... , v_{k-1}\}$ cannot be a basis for V .
I am trying to prove this by contradiction.
- Suppose $\{v_1, v_2, ... , v_{k-1}\}$ is a basis for V. Then span of this set is equal to V. So any vector in V can be written as a unique linear combination as the vectors in this set.
- $v_k$ belongs to V thus $v_k= c_1v_1 + c_2v_2 + ... + c_{k-1}v_{k-1}$.
I'm stuck now, how can I use linear independence to show a contradiction?
As you have already assumed that $\{v_1,...,v_k\}$ is a basis for $V$ so $v_k$ can not written as linear combination of $\{v_1,..,v_{k-1} \}$.Hence Contradiction.