A1(Ax)ᵢ = Σⱼ Aᵢⱼxⱼ
action of a matrix (Proof 68)
A2u·v = |u||v| cos θ
dot product and angle (Proof 156)
A3u·v = u₁v₁ + u₂v₂
dot product in components (Proof 156)
A4|u| = √(u₁² + u₂²)
magnitude of a vector (Proof 21)
A5sin²θ + cos²θ = 1
Pythagorean identity (Proof 33)
A60<θ<π ⟹ sin θ > 0
sign of the sine
A7height = |v| sin θ
sin in a right triangle (Proof 32)
A8triangle area = ½ base × height
area of a triangle (Proof 150)
A9non-overlapping ⟹ areas add
additivity of area
A10congruent figures have equal area
area of congruent figures
A11three equal sides ⟹ congruent
SSS congruence (Euclid I.8)
A12a diagonal splits a quadrilateral in two
decomposition of a convex quadrilateral (Proof 150)
A13opposite sides of a parallelogram are equal
Euclid I.34
A14segments and points have area 0
meaning of area
A15√(t²) = |t|
square root and absolute value (Proof 97)
A16x ≥ 0 ⟹ (√x)² = x
definition of the square root
A17u ≠ 0 ⟹ |u| > 0
magnitude and 0
A180<a, 0<b ⟹ 0 < ab
order axiom (positive multiplication)
A19a(b + c) = ab + ac
distributive law
A20(a − b)c = ac − bc
distributive law (subtraction form)
A21ab = ba, (ab)c = a(bc)
commutativity and associativity of multiplication
A22a + b = b + a
commutativity of addition
A23(a+b)+c = a+(b+c)
associativity of addition
A24a + (−a) = 0, a + 0 = a
additive inverse and identity
A25a × 1 = a, a × 0 = 0
multiplicative identity and 0
A26|−a| = |a|, 0<a ⟹ |a|=a
rules of absolute value
A27AB means B, then A
composition and matrix product (Proof 68)
Ea = b ⟹ apply the same operation to both sides
properties of equality
Nreal computation is determinate
entrywise computation
Nothing outside this list may be cited.