# Span and Independence of Quotient Vector Space

Let $$V/U$$ be a quotient vector space with the standard equivalence relation $$v_1 \sim v_2$$ iff $$v_2 - v_1 \in U$$.

Let $$(v_i \mid i \in I)$$ be a collection of vectors in $$V$$ indexed by $$I$$. Then $$([v_i]\mid i \in I)$$ is a collection to vectors in $$V/U$$.

Claim 1: $$[x] \in \operatorname{span}([v_i] \mid i \in I)$$ is equivalent to writing $$x = \lambda_1v_{i_1} + \cdots + \lambda_n v_{i_n} + u$$ for some collection of indices $$(i_k)$$, scalars $$\lambda$$, and $$u \in U$$.

Claim 2: $$([v_i] \mid i \in I)$$ is linear independent is equivalent to any relationship following $$0 = \lambda_1 v_{i_1} + \cdots + \lambda_n v_{i_n} + u$$ necessitates that $$\lambda_k = 0$$.

I'm unsure why both of these claims are true and what role the quotient space plays.

• They're true because if you map these relations in the quotient space $V/U$, the vector $u$ maps to $[0]$. – Bernard Mar 9 at 21:12