# Linear algebra proof

## Homework Statement

Prove the following: Let V be a vector space and assume there is an integer n such that if (v1, . . . , vk) is a linearly independent sequence from V then k ≤ n. Prove is (v1, . . . , vk) is a maximal linearly independent sequence from V then (v1, . . . , vk) spans V and is therefore a basis.

## The Attempt at a Solution

If v_1,.....,v_k\$spans V then all vectors in V are generated by some linear combination of v_1,...,v_k. It's clearly seen that we can generate any vector in the sequence by setting the constant of the desired vector to 1 and the others to 0. Hence the sequence is maximal linearly independent, adding another vector will provoke a dependency. The dependency didn't exist before the new vector was added. This implies that the added vector can be written as a linear combination of the other vectors. If we do this for every remaining vector in V, then all vectors can be written as a linear combination of the given sequence and therefore it spans V. The sequence is linearly independent and it spans V, so it's a basis.

is that correct?

haruspex
Homework Helper
Gold Member
You are asked to prove it spans V. Starting with "if v1..vk spans V" is not helpful.

You are asked to prove it spans V. Starting with "if v1..vk spans V" is not helpful.
So if I delete the first sentence, it would look better? because clearly that's what I want to prove.

haruspex