Alphabeta MathMatrices and Change of Basis: Examples and Counterexamples
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 7 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Example
A matrix represents a map F²→ F³ by its images of the standard basis vectors
Example
The same operator has two different matrices in two ordered bases
Example
The quarter-turn (x,y)↦(-y,x) on ℝ² has matrix beginpmatrix0&-11&0 endpmatrix and square -I₂
Example
Two explicit 2 by 2 matrices do not commute
Example
A nonzero square-zero matrix is not similar to any diagonal matrix
Example
Changing both domain and codomain bases of a map F²→ F³ uses both sides of the formula
Example