Quotient by the diagonal line

Goal and scope

R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal.

Full equivalence and representative-independent operations for a finite-dimensional real quotient vector space; no topological quotient is considered.

Givens and symbols

• calc

• G1 — V=R2, W={t(1,1):t in R}; [v]=v+W and q(x,y)=y-x.

• [v] — whole coset
• q — coset label y-x
• b — basis representative (0,1)

Rules and sources

• calc · prerequisite — Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations.
• mko_definition · cited — Vectors v,u name the same coset exactly when v−u lies in W. Quotient elements are cosets, not unique representatives. Check representative independence of [v]+[u]=[v+u] and scalar*[v]=[scalar*v]. In finite dimension dim(V/W)=dim(V)−dim(W), and the zero coset is W. · mko-quotient-vector-space

Main derivation

W=\{(t,t):t\in\mathbb R\}

0\in W,\quad(s,s)+(t,t)=(s+t,s+t),\quad\lambda(t,t)=(\lambda t,\lambda t)

v=(x,y),\quad u=(a,b)

v-u\in W\iff y-b=x-a\iff q(v)=q(u)

v'=v+w_1,\quad u'=u+w_2,\quad w_1,w_2\in W

(v'+u')-(v+u)=w_1+w_2\in W

v'=v+w,\quad w\in W,\quad\lambda\in\mathbb R

\lambda v'-\lambda v=\lambda w\in W

[0]=0+W

[0]=W,\quad[(1,1)]=[(0,0)]

v=(x,y),\quad b=(0,1)

v-(y-x)b=(x,x)\in W\Rightarrow[v]=(y-x)[b]

\lambda[b]=[0]

(0,\lambda)\in W\iff\lambda=0

\{[b]\}\text{ independent and spanning}

\dim(V/W)=1=2-1=\dim V-\dim W

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S01, S06, S07 — This is an instance certificate, not a general abstract proof.
• S07 · premise_audit · G1, S01, S04, S05 — A nonzero vector need not have a nonzero coset; b outside W is checked here.
• S06 · premise_audit · G1, S02, S04 — This uses the well-defined quotient scalar operation, not numerical division.
• S05 · premise_audit · G1, S01, S02 — The representative vectors differ, but their difference is in W, so their cosets agree.
• S04 · premise_audit · G1, S01, S02 — Zero and negative scalars are included without extra restrictions.
• S03 · premise_audit · G1, S01, S02 — This uses additive closure of W, not preferred-representative tests.
• S02 · inverse_equivalence · G1, S01 — The equivalence covers all real coordinates, not finite membership tests.
• S01 · premise_audit · G1 — Subspace qualification precedes the vector quotient.

Error controls and negative controls

Zero type

The quotient zero is W itself, not a unique representative.

Both well-defined operations

Addition and scalar multiplication each require universal closure of W.

• The zero coset contains only (0,0). → reject — It is W, containing all (t,t).
• [(1,0)] differs from [(2,1)]. → reject — Their difference (1,1) is in W.
• Every nonzero vector represents a nonzero coset. → reject — (1,1) is nonzero but lies in W.

Result and limitations

R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 771c4e584bb1897c89adcfcd5b4e225c158db4a9c6492ad3ea24f5772b21d017
• original source SHA-256: f3cefaafea5ecc2a3034c11fb42fd17ee54ad688f67e38f2f43a3a14ed5f3351
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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