# Concept derivation guides 299 concept teaching guides, separate from the 36,000 individual question derivations. [Search all questions & derivations →](/en/explore/) [← Previous](/en/derivations/?page=7#results)[1](/en/derivations/?page=1#results) … [7](/en/derivations/?page=7#results) 8 [9](/en/derivations/?page=9#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=9#results) 299 results · Page 8 of 13 Concept guide ## [A complete coloring and a nonoptimal greedy trace](/en/derivations/derivation-mko-graph-coloring-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Audit a proper coloring edgewise and show an upper bound need not be optimal. Graph coloring derivation-mko-graph-coloring-01 Concept guide ## [Universal closure of a line through zero](/en/derivations/derivation-mko-vector-subspace-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not. Real vector subspace derivation-mko-vector-subspace-01 Concept guide ## [Closure and interior in a three-point chain topology](/en/derivations/derivation-mko-closure-and-interior-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Find the closure and interior of A={b}. Topological closure and interior derivation-mko-closure-and-interior-01 Concept guide ## [Selecting a subcover by original indices](/en/derivations/derivation-mko-open-cover-and-subcover-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Audit a supplied open cover of K={a,b} and its subcover indices. Open cover and subcover derivation-mko-open-cover-and-subcover-01 Concept guide ## [Atomwise tails and a null exception](/en/derivations/derivation-mko-almost-sure-convergence-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Give a convergence or failure certificate for every atom. Almost-sure convergence derivation-mko-almost-sure-convergence-01 Concept guide ## [Maximal versus maximum in P4](/en/derivations/derivation-mko-graph-matching-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Audit selected edges and endpoints, then distinguish matching properties. Graph matching derivation-mko-graph-matching-01 Concept guide ## [Counting phi(12) representative by representative](/en/derivations/derivation-mko-euler-totient-function-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) phi(12)=4, phi(1)=1, and here phi(3*4)=phi(3)*phi(4). Euler totient function derivation-mko-euler-totient-function-01 Concept guide ## [Tracing 1011 through a complete DFA](/en/derivations/derivation-mko-finite-automaton-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) The specified DFA rejects 1011 and accepts epsilon. Deterministic finite automaton derivation-mko-finite-automaton-01 Concept guide ## [Prove a basis before reading dimension](/en/derivations/derivation-mko-vector-space-dimension-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) B=((1,1),(1,-1)) is an ordered basis of R2, so dim R2=2. Dimension of a finite-dimensional vector space derivation-mko-vector-space-dimension-01 Concept guide ## [The full kernel of a two-by-two map](/en/derivations/derivation-mko-linear-map-kernel-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) The kernel of T(x,y)=(x+2y,2x+4y) is span{(-2,1)}. Kernel of a linear map derivation-mko-linear-map-kernel-01 Concept guide ## [The full image from consistency](/en/derivations/derivation-mko-linear-map-image-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not. Image of a linear map derivation-mko-linear-map-image-01 Concept guide ## [Exact boundary comparison with the complement](/en/derivations/derivation-mko-topological-boundary-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Compute the boundaries of A={b} and its complement in a chain topology. Topological boundary derivation-mko-topological-boundary-01 Concept guide ## [A universal compactness certificate for a finite topology](/en/derivations/derivation-mko-compact-topological-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Prove every open cover of the three-point chain topology has a finite subcover. Compact topological space derivation-mko-compact-topological-space-01 Concept guide ## [Bayes reversal in a four-atom test model](/en/derivations/derivation-mko-bayes-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Compute priors, likelihoods and posteriors from a complete joint law. Bayes theorem derivation-mko-bayes-theorem-01 Concept guide ## [A parity invariant for arbitrary word length](/en/derivations/derivation-mko-regular-language-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) All words over {0,1} with an even number of 1s form a regular language. Regular language derivation-mko-regular-language-01 Concept guide ## [Full RREF and rank bounds](/en/derivations/derivation-mko-matrix-rank-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) A=[[1,2,3],[2,4,6],[1,1,1]] has row and column rank 2. Matrix rank derivation-mko-matrix-rank-01 Concept guide ## [Pointwise tail certificates on a finite domain](/en/derivations/derivation-mko-pointwise-convergence-of-functions-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Prove pointwise convergence of f_n(p)=1+2/(n+1), f_n(q)=-2-3/(n+1) to f. Pointwise convergence of functions derivation-mko-pointwise-convergence-of-functions-01 Concept guide ## [Complete tails for geometric and telescoping series](/en/derivations/derivation-mko-infinite-series-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Prove convergence of two series with exact partial sums and remainders. Infinite series derivation-mko-infinite-series-01 Concept guide ## [A complete model with two parameter values and four sample points](/en/derivations/derivation-mko-parametric-statistical-model-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Validate every parameter law and distinguish likelihood from posterior. Parametric statistical model derivation-mko-parametric-statistical-model-01 Concept guide ## [Excluding every nontrivial clopen set](/en/derivations/derivation-mko-connected-topological-space-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Decide connectedness of a three-point chain topology. Connected topological space derivation-mko-connected-topological-space-01 Concept guide ## [All eight edge-subset candidates on a three-edge graph](/en/derivations/derivation-mko-maximum-matching-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Enumerate all2³ subsets for the maximum matching and all witnesses. Maximum matching derivation-mko-maximum-matching-01 Concept guide ## [The normalized inverse of 17 modulo 43](/en/derivations/derivation-mko-modular-inverse-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) The unique inverse residue class of 17 modulo 43 is represented by 38; 17*38=15*43+1. Modular inverse derivation-mko-modular-inverse-01 Concept guide ## [Rank and nullity for three source coordinates](/en/derivations/derivation-mko-rank-nullity-theorem-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) For T:R3 to R2, A=[[1,2,3],[2,4,6]] has rank 1 and nullity 2, with 3=1+2. Rank-nullity theorem derivation-mko-rank-nullity-theorem-01 Concept guide ## [Exact supremum error on a three-point domain](/en/derivations/derivation-mko-uniform-convergence-of-functions-01/?from=%2Fen%2Fderivations%2F%3Fpage%3D8) Prove g_n(a)=1/(n+2),g_n(b)=-4/(n+2),g_n(c)=2/(n+2) converges uniformly to zero. Uniform convergence of functions derivation-mko-uniform-convergence-of-functions-01 [← Previous](/en/derivations/?page=7#results)[1](/en/derivations/?page=1#results) … [7](/en/derivations/?page=7#results) 8 [9](/en/derivations/?page=9#results) … [13](/en/derivations/?page=13#results)[Next →](/en/derivations/?page=9#results) 299 results · Page 8 of 13