A Group is a non-empty set G with a binary operation * satisfying:
(1) Closure: a*b ∈ G for all a,b ∈ G
(2) Associativity: (a*b)*c = a*(b*c)
(3) Identity: ∃ e ∈ G such that a*e = e*a = a
(4) Inverse: For each a, ∃ a⁻¹ such that a*a⁻¹ = e
Example: (ℤ, +) – integers under addition; identity = 0, inverse of n is -n.

The Rank of a matrix A is the maximum number of linearly independent rows (or columns) in the matrix. Equivalently, it is the order of the largest non-zero minor.
For matrix [[1,2,3],[4,5,6],[5,7,9]]: Row reduce to find rank = 2 (rows are dependent).
Rank determines consistency of linear systems: rank(A) = rank(augmented) → consistent.

Scalar Triple Product: [a, b, c] = a·(b×c). It gives the volume of the parallelepiped formed by three vectors. If [a,b,c] = 0, vectors are coplanar.

Vector Triple Product: a×(b×c) = (a·c)b − (a·b)c. The result is a vector lying in the plane of b and c.

An ellipse is the locus of a point P such that the sum of its distances from two fixed points (foci F₁ and F₂) is constant: PF₁ + PF₂ = 2a.
Standard Equation: x²/a² + y²/b² = 1 (a > b > 0)
where a = semi-major axis, b = semi-minor axis, c² = a²-b², eccentricity e = c/a (0 < e < 1).

Theorem: Every subgroup of a cyclic group is cyclic.

Proof: Let G = ⟨a⟩ be a cyclic group and H be a subgroup of G.
Case 1: If H = {e}, then H = ⟨e⟩ is cyclic.
Case 2: If H ≠ {e}, let m be the smallest positive integer such that aᵐ ∈ H.
Claim: H = ⟨aᵐ⟩.
Any element h ∈ H is of form aⁿ. By division algorithm: n = mq + r, 0 ≤ r < m.
Then aʳ = aⁿ·(aᵐ)⁻q ∈ H. Since r < m and m is smallest, r = 0.
So n = mq → h = (aᵐ)q ∈ ⟨aᵐ⟩. Thus H ⊆ ⟨aᵐ⟩ and ⟨aᵐ⟩ ⊆ H → H = ⟨aᵐ⟩ is cyclic. ✓

Cayley's Theorem: Every group G is isomorphic to a subgroup of the symmetric group S(G).

Proof: For each g ∈ G, define φg: G → G by φg(x) = gx (left multiplication). Then φg is a bijection. Define Φ: G → S(G) by Φ(g) = φg. Then Φ is a group homomorphism and is injective (since φg = φh implies g = h). Therefore G ≅ Φ(G) ⊆ S(G). ✓

The general equation of a circle: x² + y² + 2gx + 2fy + c = 0
Three points determine unique circle. Let the points be A(1,0), B(0,1), C(1,1).
Substituting each: 1+0+2g+0+c=0 → 2g+c=-1
0+1+0+2f+c=0 → 2f+c=-1
1+1+2g+2f+c=0 → 2g+2f+c=-2
From (1)&(2): g = f. From (3): 2g+2g+c=-2 → 4g+c=-2. With 2g+c=-1 → 2g=-1 → g=-½, f=-½, c=0.
Equation: x² + y² - x - y = 0.

For a square matrix A, a non-zero vector v is an eigenvector with eigenvalue λ if: Av = λv
To find eigenvalues: solve det(A - λI) = 0 (characteristic equation).
To find eigenvectors: solve (A - λI)v = 0 for each λ.

Example: A = [[2,1],[0,3]]. det(A-λI) = (2-λ)(3-λ) = 0 → λ₁=2, λ₂=3.
For λ=2: (A-2I)v=0 → v=[1,0]ᵀ. For λ=3: v=[1,1]ᵀ.

The angle θ between two planes a₁x+b₁y+c₁z=d₁ and a₂x+b₂y+c₂z=d₂ is the angle between their normal vectors n₁=(a₁,b₁,c₁) and n₂=(a₂,b₂,c₂):

cos θ = |n₁·n₂|/(|n₁||n₂|) = |a₁a₂+b₁b₂+c₁c₂| / √(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)

If θ = 0°, planes are parallel. If θ = 90°, planes are perpendicular (a₁a₂+b₁b₂+c₁c₂=0).

Lagrange's Theorem: If H is a subgroup of a finite group G, then |H| divides |G|.

Proof: Define relation: a ~ b if a⁻¹b ∈ H. This is an equivalence relation. The equivalence class of a = aH (left coset). All cosets have same size |H|. G is partitioned into distinct cosets, each of size |H|. So |G| = (number of cosets) × |H| → |H| divides |G|. ✓

Properties of Determinants:

1. Row/Column interchange: Interchanging two rows/columns changes sign of determinant.
2. Identical rows: If two rows/columns are identical, det = 0.
3. Scalar multiplication: Multiplying one row by k multiplies det by k.
4. Sum property: det with row = sum of two rows = sum of two determinants.
5. Row operations: Adding multiple of one row to another leaves det unchanged.
6. Zero row: If all elements of a row/column are zero, det = 0.
7. Triangular matrix: det = product of diagonal elements.
8. Transpose: det(A) = det(Aᵀ).
9. Multiplicative property: det(AB) = det(A)·det(B).

Sphere: Locus of points equidistant from a fixed point (center).
Standard equation: x² + y² + z² = r² (center at origin)
General equation: x² + y² + z² + 2ux + 2vy + 2wz + d = 0
Center = (-u, -v, -w), Radius = √(u²+v²+w²-d)

Properties:

• A sphere is determined by 4 non-coplanar points.
• Intersection of sphere with a plane is a circle.
• Two spheres intersect in a circle (common circle).
• Tangent plane at (x₁,y₁,z₁): xx₁+yy₁+zz₁ = r².
• Great circle: circle passing through center of sphere.