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.

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

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