# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=2#results)[1](/en/knowledge/?page=1#results) [2](/en/knowledge/?page=2#results) 3 [4](/en/knowledge/?page=4#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=4#results) 302 results · Page 3 of 13 Knowledge object ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A solution must satisfy two real linear constraints simultaneously. A nonzero determinant gives a unique solution; singular systems and zero rows require separate consistency checks. mko-system-linear-equations · Algebra Knowledge object ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) The conditioning event must have positive probability. In a finite model, P(A|B)=P(A∩B)/P(B), and the direction cannot be freely exchanged. mko-conditional-probability · Probability and statistics Knowledge object ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain. mko-injective · Foundations and logic Knowledge object ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages. mko-surjective · Foundations and logic Knowledge object ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage. mko-bijective · Foundations and logic Knowledge object ## [Inverse functions of bijections](/en/mko/mko-inverse-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction. mko-inverse-function · Algebra Knowledge object ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight. mko-random-variable · Probability and statistics Knowledge object ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence. mko-expectation · Probability and statistics Knowledge object ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it. mko-variance · Probability and statistics Knowledge object ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend. mko-order-relation · Foundations and logic Knowledge object ## [Quadratic functions in vertex form](/en/mko/mko-quadratic-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign. mko-quadratic-function · Algebra Knowledge object ## [Average rate of change between distinct inputs](/en/mko/mko-rate-of-change/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative. mko-rate-of-change · Analysis Knowledge object ## [Euclidean distance in an equal-unit orthonormal frame](/en/mko/mko-distance-formula/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences. mko-distance-formula · Geometry Knowledge object ## [Circle boundaries and zero-radius degeneracy](/en/mko/mko-circle/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton. mko-circle · Geometry Knowledge object ## [Exact area of half-open rectangles and unit-cell unions](/en/mko/mko-area/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases. mko-area · Geometry Knowledge object ## [Descriptive summaries of nonempty observation lists](/en/mko/mko-descriptive-statistics/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities. mko-descriptive-statistics · Probability and statistics Knowledge object ## [Finite matrices: dimensions, indexing, and transpose](/en/mko/mko-matrix/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Matrices organize entries by rows and columns; this pulse uses nonempty finite real matrices and exact integer-array instances. mko-matrix · Algebra Knowledge object ## [Matrix-represented linear maps on the real plane](/en/mko/mko-linear-transformation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Use the explicit R²→R² matrix model for additivity and scalar compatibility; an affine expression with nonzero translation is not linear. mko-linear-transformation · Algebra Knowledge object ## [Determinants of real 2×2 matrices](/en/mko/mko-determinant/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) This entry covers ad−bc, invertibility over R, product determinants, and scalar scaling for 2×2 real matrices. mko-determinant · Algebra Knowledge object ## [Real eigenvalues and nonzero eigenvectors of 2×2 matrices](/en/mko/mko-eigenvalue/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) An eigenvector must be nonzero, whereas a candidate-scalar kernel includes zero; a real 2×2 matrix may have no real eigenvalues. mko-eigenvalue · Algebra Knowledge object ## [Congruence of nondegenerate triangles with explicit correspondence](/en/mko/mko-congruence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) In an equal-unit orthonormal plane, this slice compares all three corresponding side lengths of labeled nondegenerate triangles and allows reflection. mko-congruence · Geometry Knowledge object ## [Labeled triangle similarity with a specified positive scale](/en/mko/mko-similarity/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Specify the positive length scale k from A to B; squared lengths and a given area scale by k². A negative coordinate coefficient is not a negative length scale. mko-similarity · Geometry Knowledge object ## [Plane-coordinate isometries and uniform scaling](/en/mko/mko-geometric-transformations/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) Translations, counterclockwise quarter-turn rotations about the origin, and specified line reflections are plane isometries; uniform scaling distinguishes zero, ±1, and other coefficients. mko-geometric-transformations · Geometry Knowledge object ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fknowledge%2F%3Fpage%3D3) A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model. mko-sequence-limit · Analysis [← Previous](/en/knowledge/?page=2#results)[1](/en/knowledge/?page=1#results) [2](/en/knowledge/?page=2#results) 3 [4](/en/knowledge/?page=4#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=4#results) 302 results · Page 3 of 13