derivation-mko-direct-sum-01 · ocme-new142-reader-adoption-20261021
R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified.
A finite-dimensional two-summand internal sum, with many-summand and external-carrier boundaries.
calcG1 — V=R3; e1=(1,0,0),e2=(0,1,0),e3=(0,0,1); U=span{e1}, W=span{e2,e3}.U,W — internal subspaces\Phi — addition map from external to internal sumcalc · 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 — Use complete bases and exact ranks in a finite-dimensional real ambient space. U∩W={0} ensures uniqueness in U+W, while V=U⊕W also needs U+W=V. For three or more subspaces pairwise zero intersections do not replace global uniqueness; an external direct sum uses a new product-type carrier. · mko-direct-sumBefore
U=\operatorname{span}\{e_1\},\quad W=\operatorname{span}\{e_2,e_3\}After
U=\{(a,0,0):a\in\mathbb R\},\quad W=\{(0,b,c):b,c\in\mathbb R\}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
Both subspaces are fully parameterized; each generating family is independent by coordinate comparison.
Before
v\in U\cap W
After
v=(a,0,0)=(0,b,c)\Rightarrow a=b=c=0
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Comparing every coordinate makes the intersection the zero subspace.
Before
(x,y,z)\in V
After
(x,y,z)=(x,0,0)+(0,y,z)\in U+W
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Every ambient vector has two valid summands, so U+W=V.
Before
u+w=u'+w',\quad u,u'\in U,\quad w,w'\in W
After
u-u'=w'-w\in U\cap W=\{0\}Conditions: none
Rule: calc · Uses: G1, S01, S02
Leaf basis: calc
The difference of two decompositions lies in the intersection and therefore vanishes.
Before
u-u'=0,\quad w'-w=0
After
u=u',\quad w=w';\qquad V=U\oplus W
Conditions: none
Rule: mko_definition · Uses: G1, S03, S04
Leaf basis: mko_definition
Uniqueness combines with coverage to certify the internal direct sum of V.
Before
[e_1\ e_2\ e_3]=I_3
After
\operatorname{rank}[e_1\ e_2\ e_3]=3,\quad1+2=3Conditions: none
Rule: calc · Uses: G1, S01, S05
Leaf basis: calc
The full concatenated basis matrix is I3; its three pivots recheck independence and coverage.
Before
L_1=\mathbb R(1,0),\ L_2=\mathbb R(0,1),\ L_3=\mathbb R(1,1)
After
L_i\cap L_j=\{0\}\ (i\ne j),\quad(1,0)+(0,1)-(1,1)=0Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
Three R2 lines have pairwise zero intersections yet a nontrivial global relation, so the three-way sum is not direct.
Before
U\times W=\{(u,w):u\in U,w\in W\}After
\Phi:U\times W\to V,\quad\Phi(u,w)=u+w\text{ bijective}Conditions: none
Rule: calc · Uses: G1, S03, S04, S05
Leaf basis: calc
External-sum elements are ordered pairs; coverage gives surjectivity and uniqueness gives injectivity here.
No substeps in this guide.
S08 · premise_audit · G1, S03, S04, S05 — A bijection does not make ordered pairs literally equal to ambient vectors.S07 · premise_audit · G1 — Three nonzero summands add to zero, directly refuting uniqueness.S06 · premise_audit · G1, S01, S05 — The rank certificate concerns the same U,W, not arbitrary dimensions.S05 · premise_audit · G1, S03, S04 — Zero intersection alone gives uniqueness within U+W; S03 cannot be omitted.S04 · premise_audit · G1, S01, S02 — Closure under differences is needed to place each side in its respective subspace.S03 · premise_audit · G1, S01 — Zero intersection and coverage are separate; this step establishes coverage.S02 · premise_audit · G1, S01 — Zero belongs to both, so the intersection is not empty.S01 · premise_audit · G1 — Omitting e3 from W would invalidate coverage.Zero intersection gives uniqueness; U+W=V gives coverage. Both are needed.
A many-summand direct sum requires global independence of all basis columns.
reject — span{e1}+span{e2} misses e3.reject — The displayed lines have a nontrivial three-term relation.reject — The external carrier contains (u,w); Phi sends it to u+w.R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified.
1 declared checks for this guide; only the listed expressions are covered.
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