Rank and nullity for three source coordinates

Goal and scope

For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2.

Complete kernel, image and source-dimension certificates for one exact matrix, not a general proof.

Givens and symbols

• calc

• G1 — A is 2-by-3, [[1,2,3],[2,4,6]], T(x,y,z)=A(x,y,z)^T, source R3 and target R2.

• u,v — kernel-basis vectors
• s — nonkernel coefficient

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 — An explicit m-by-n matrix represents T:R^n→R^m. Exact RREF, a full kernel basis and column-space rank verify n=nullity+rank. Nullity is kernel dimension, not the number of solutions; sampled vectors do not establish the identity in general. · mko-rank-nullity-theorem

Main derivation

A\in\mathbb R^{2\times3},\quad T:\mathbb R^3\to\mathbb R^2

\dim(\operatorname{domain}T)=3

\begin{pmatrix}1&2&3\\2&4&6\end{pmatrix}

R_2\leftarrow R_2-2R_1:\quad\begin{pmatrix}1&2&3\\0&0&0\end{pmatrix}

T(x,y,z)=(x+2y+3z)(1,2)

\operatorname{im}T=\operatorname{span}\{(1,2)\},\quad T(s,0,0)=s(1,2)

T(x,y,z)=0

x=-2y-3z,\quad(x,y,z)=y(-2,1,0)+z(-3,0,1)

u=(-2,1,0),\quad v=(-3,0,1)

Au=Av=0,\quad\alpha u+\beta v=0\Rightarrow\alpha=\beta=0

\mathcal B_{\ker}=(u,v),\quad\mathcal B_{\operatorname{im}}=((1,2))

\operatorname{nullity}T=2,\quad\operatorname{rank}T=1

\dim\mathbb R^3=3,\quad\operatorname{nullity}T=2,\quad\operatorname{rank}T=1

3=2+1=\dim\ker T+\dim\operatorname{im}T

(x,y,z)=s(1,0,0)+y(-2,1,0)+z(-3,0,1)

s=x+2y+3z

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S03, S04, S05 — Coordinates two and three fix y,z, then the first fixes s, proving uniqueness.
• S07 · premise_audit · G1, S01, S06 — This is not a numerical coincidence; S03 and S05 give full image and kernel bases.
• S06 · premise_audit · G1, S03, S04, S05 — Real kernel parameters give infinitely many elements but dimension 2.
• S05 · premise_audit · G1, S04 — The basis is directly qualified rather than merely counting parameters.
• S04 · inverse_equivalence · G1, S02 — y,z are arbitrary real numbers, not a finite grid.
• S03 · premise_audit · G1, S02 — Nonzero (1,2) is independent and gives a one-vector image basis.
• S02 · premise_audit · G1, S01 — The next step directly certifies the column space rather than guessing rank.
• S01 · premise_audit · G1 — Shape fixes map direction; target dimension cannot replace source dimension.

Error controls and negative controls

Map direction

An m-by-n matrix has source R^n, here n=3.

Dimension not cardinality

Nullity counts basis vectors, not infinitely many solutions.

• A 2-by-3 matrix has source dimension 2. → reject — Its three-coordinate input belongs to R3.
• An infinite kernel means infinite nullity. → reject — Its full two-vector basis gives nullity 2.
• A few tested kernel vectors prove the full kernel. → reject — The universal solution S04 and independence S05 are required.

Result and limitations

For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 1e139ae9f1da30da2e69441c4c7cbb917eb85b0291ab7065aaec2ef3b2d88dec
• original source SHA-256: 8719bc547d83d493161e4c36002b5c8bd70cec444fe62f2391bb69b5b395450a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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