| Method |
Description |
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dataProviderForAdditiveInverse ( ) |
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dataProviderForCrossProduct ( ) |
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dataProviderForCrossProductThreeVectors ( ) |
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dataProviderForDotProductZero ( ) |
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dataProviderForOuterProduct ( ) |
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dataProviderForPerpendicularIdentity ( ) |
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dataProviderForProjPerp ( ) |
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dataProviderForSingleVector ( ) |
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dataProviderForTwoVectors ( ) |
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testAdditivityInTheScalarForScalarMultiplication ( array $A ) |
Axiom: (c + d)A = cA + dA
Additivity in the scalar |
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testAdditivityInTheVectorForScalarMultiplication ( array $A, array $B ) |
Axiom: c(A + B) = cA + cB
Additivity in the vector |
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testCrossProductDistributivity ( array $A, array $B, array $C ) |
Axiom: A x (B + C) = (A x B) + (A x C)
Cross product distributivity |
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testCrossProductInnerProductWithEitherVectorIsZero ( array $A, array $B ) |
Axiom: (A x B) ⋅ A = 0
Axiom: (A x B) ⋅ B = 0
Dot product of either vector with the cross product is always zero. |
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testCrossProductLagrangeFormula ( array $A, array $B, array $C ) |
Axiom: A x (B x C) = (A ⋅ C)B - (A ⋅ B)C
Lagrange's formula |
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testCrossProductPropertyOfZero ( array $A, array $_ ) |
Axiom: A x 0 = 0
Cross product property of 0 |
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testCrossProductVolumeProperty ( array $A, array $B, array $C ) |
Axiom: A ⋅ (B x C) = (A x B) ⋅ C
Cross product volumn property |
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testDotProductCommutative ( array $A, array $B ) |
Axiom: A⋅B = B⋅A
Dot product is commutative |
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testDotProductZero ( array $A, array $zero ) |
Axiom: 0⋅A = A⋅0 = 0
Dot product of a vector and zero is zero. |
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testL2NormLessThanL1NormLessThanSqrtNL2Norm ( array $V ) |
Axiom: |x|₂ ≤ |x|₁ ≤ √n |x|₂
l²-norm is less than equal to l₁-norm which is less than equal to sqrt n * l²-norm. |
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testMaxNormLessThanEqualL1NormLessThanEqualSqrtNMaxNorm ( array $V ) |
Axiom: |x|∞ ≤ |x|₁ ≤ √n |x|∞
Max norm is less than equal to l₁-norm which is less than equal to sqrt n * max norm. |
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testMaxNormLessThtanEQualL2NormLessThanEqualSqrtNMaxNorm ( array $V ) |
Axiom: |x|∞ ≤ |x|₂ ≤ √n |x|∞
Max norm is less than equal to l₂-norm which is less than equal to sqrt n * max norm. |
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testOuterProductIsMatrixMultiplicationOfAAndBTranspose ( array $A, array $B ) |
Axiom: A⨂B = ABᵀ
Outer product is the same as matrix multiplication of A and transpose of B |
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testPerpDotProdcutSwapOperandsChangeSign ( array $A ) |
Axiom: A⋅A⊥ = -A⊥⋅A
Swapping operands changes the sign of the perp dot product |
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testPerpDotProductZero ( array $A ) |
Axiom: A⊥⋅A = 0
Perp dot product with itself will be zero. |
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testPerpendicularDotProduct ( array $A ) |
Axiom: A⋅A⊥ = 0
Vector dot product with a vector perpendicular to it will be zero. |
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testProjPerpDotProductEqualsZero ( array $A, array $B ) |
Axiom: projᵇA ⋅ perpᵇA = 0
Dot product of proj and perp of A on B is 0 |
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testProjPerpPerpDotProductEqualsProductOfLengths ( array $A, array $B ) |
Axiom: |projᵇA⊥ ⋅ perpᵇA| = |projᵇA| |perpᵇA|
Absolute value of proj and perp dot product equals product of their lengths. |
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testProjPerpSumEqualsA ( array $A, array $B ) |
Axiom: projᵇA + perpᵇA = A
Sum of the proj and perp of A on B equals A |
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testProjPerpSumOfSquares ( array $A, array $B ) |
Axiom: |projᵇA|² + |perpᵇA|² = |A|²
Sum of squared lengths of proj and perp equals squared length of A |
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testReverseCrossProduct ( array $A, array $B ) |
Axiom: A x B = -(B x A)
Anticommutivity: Reverse order cross product results in a negative cross product |
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testScalarMultiplyNegativeOneIdentity ( array $A, array $R ) |
Axiom: -1A = -A
Additive inverse |
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testScalarMultiplyOneIdentity ( array $A ) |
Axiom: 1A = A
Multiplying (scaling) by 1 does not change the vector |
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testScalarMultiplyZeroIdentity ( array $A ) |
Axiom: 0A = 0
Multiplying (scaling) by 0 gives the zero vector |
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