derivation-mko-dual-vector-space-01 · ocme-new142-reader-adoption-20261021
For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis.
Explicit R2 functionals, universal linearity and four pairings; no canonical identification without extra structure.
calcG1 — V=R2, B=((1,1),(1,-1)), P=[[1,1],[1,-1]] is the basis-column matrix.V^* — real linear dual\phi_i — dual-basis functionalP — primal basis columnscalc · 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 — This slice uses finite-dimensional V=R^n. A row covector gives a symbolic linearity certificate, and a dual basis is checked by phi_i(v_j)=delta_ij. A primal vector and a dual functional are distinct; without extra structure no canonical identification V→V* is claimed. · mko-dual-vector-spaceBefore
\alpha b_1+\beta b_2=(\alpha+\beta,\alpha-\beta)
After
(x,y)=\frac{x+y}{2}b_1+\frac{x-y}{2}b_2\text{ uniquely}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
Every vector has this unique representation, qualifying B as a basis first.
Before
\phi_1(x,y)=\frac{x+y}{2},\quad\phi_2(x,y)=\frac{x-y}{2}After
\phi_i(au+bv)=a\phi_i(u)+b\phi_i(v)\quad(i=1,2)
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Each fixed row covector is linear by distributivity for all real a,b.
Before
b_1=(1,1)
After
\phi_1(b_1)=1,\quad\phi_2(b_1)=0
Conditions: none
Rule: calc · Uses: G1, S02
Leaf basis: calc
Evaluating the first basis vector checks the first delta column.
Before
b_2=(1,-1)
After
\phi_1(b_2)=0,\quad\phi_2(b_2)=1
Conditions: none
Rule: calc · Uses: G1, S02, S03
Leaf basis: calc
Evaluating the second vector completes the four-pairing matrix I2.
Before
\ell\in V^*,\quad v=\alpha b_1+\beta b_2
After
\ell(v)=\alpha\ell(b_1)+\beta\ell(b_2)=\ell(b_1)\phi_1(v)+\ell(b_2)\phi_2(v)
Conditions: none
Rule: calc · Uses: G1, S01, S02, S03, S04
Leaf basis: calc
An arbitrary linear functional is determined by its basis values, so the two phi functionals span V*.
Before
a\phi_1+b\phi_2=0\text{ as functions}After
\text{evaluate at }b_1,b_2:\quad a=b=0Conditions: none
Rule: mko_definition · Uses: G1, S03, S04, S05
Leaf basis: mko_definition
Evaluation directly proves independence, completing the dual basis with spanning.
Before
[v]_{\rm std}=P[v]_BAfter
[v]_B=P^{-1}[v]_{\rm std},\quad P^{-1}=\frac12\begin{pmatrix}1&1\\1&-1\end{pmatrix}Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Primal coordinates use the inverse; P^2=2I exactly verifies it.
Before
\ell(x,y)=3x+y,\quad[\ell]_{\rm std}=(3,1)After
[\ell]_B=(3,1)P=(4,2),\quad\ell(v)=4\alpha+2\beta
Conditions: none
Rule: calc · Uses: G1, S01, S07
Leaf basis: calc
The covector row multiplies P rather than P inverse, opposite to primal coordinate change.
No substeps in this guide.
S08 · premise_audit · G1, S01, S07 — Substitution x=alpha+beta,y=alpha-beta gives 4alpha+2beta, checking the same functional.S07 · premise_audit · G1, S01 — P and P inverse cannot be interchanged even though they differ by a scalar here.S06 · premise_audit · G1, S03, S04, S05 — The zero functional vanishes on all inputs, so in particular on both basis vectors.S05 · premise_audit · G1, S01, S02, S03, S04 — Ell is arbitrary, not a few sampled functions.S04 · premise_audit · G1, S02, S03 — phi2(b2)=(1-(-1))/2=1, with the minus sign retained.S03 · premise_audit · G1, S02 — (1-1)/2 is zero; both entries cannot be one.S02 · premise_audit · G1, S01 — phi_i returns a real number; a functional is not a primal vector.S01 · premise_audit · G1 — Division by 2 is valid; the dual basis being constructed is not used to qualify B.Check all delta_ij pairings, not only diagonal entries.
V* is a function space, and row coordinates require a chosen basis.
reject — One is a vector, the other a function V to R.reject — (1+(-1))/2=0.reject — Here ell's B-values are (4,2), not (2,1).For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis.
2 declared checks for this guide; only the listed expressions are covered.
b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f284271845add69df42d2cff4d06bc73a7247e856826b15e73926becaccedbd91284271845add69df42d2cff4d06bc73a7247e856826b15e73926becaccedbd91SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEWSame-source full JSON (formulas, ASTs, controls and revision log)