Mathematical Concepts and Formulas, Cheat Sheet of Algebra

Download Mathematical Concepts and Formulas and more Cheat Sheet Algebra in PDF only on Docsity! 1 Complex Numbers Principal argument is −𝜋 < 𝜃 ≤ 𝜋 |𝑧|2 = 𝑧𝑧; 𝑧 + 𝑤 = 𝑧 + 𝑤; 𝑧𝑤 = 𝑧 𝑤; 𝑧 + 𝑧 = 2𝑅𝑒(𝑧); 𝑧 − 𝑧 = 2𝑖 𝐼𝑚(𝑧) Product of moduli = modulus of product (same for division) Nth roots of unity sum to 0, product is ± 1 (odd/even n). Calculate roots of z by writing in exponential form, then dividing power {𝑧 ∈ ℂ | 𝑅𝑒(ei𝜃𝑧) = 𝑟} is straight line (same for Im) for some 𝜃, 𝑟 ∈ ℝ cos 𝑧 = 1 2 (𝑒𝑖𝑧 + 𝑒−𝑖𝑧); sin 𝑧 = 1 2i (𝑒𝑖𝑧-𝑒−𝑖𝑧) Tricks: write in component/polar/exponential form; compare Re and Im parts; multiply by 𝑒𝑖𝑧; find trig functions 2 Integers and Congruences Unique 𝑞, 𝑟 ∈ ℤ with 0 ≤ 𝑟 < 𝑏 for every 𝑎 ∈ ℤ, 𝑏 ∈ ℕ in 𝑎 = 𝑞𝑏 + 𝑟 If c a common divisor of a and b, then c divides any lin comb of a, b In 𝑎 = 𝑞𝑏 + 𝑟, gcd(𝑎, 𝑏) = gcd(𝑏, 𝑟) GCD can always be written as a linear comb (v helpful, esp if gcd 1) Congruence Equations To solve 𝑎𝑥 ≡ 𝑏 (𝑚𝑜𝑑 𝑚): 1. Find 𝑑 = gcd(𝑎, 𝑚) (Euclid’s alg) 2. Check 𝑑 | 𝑏 (otherwise no sols) 3. Write d as a linear comb of a and m (Euclid’s alg backwards) 4. Factor the linear comb out of b 5. Can then remove m part, and then cancel a part 6. Solutions are sols mod 𝑚 𝑑 Chinese Remainder Theorem To solve simultaneously 𝑥 = 𝑐1(𝑚𝑜𝑑 𝑚1) and 𝑥 = 𝑐2(𝑚𝑜𝑑 𝑚2): 1. Find 𝑑 = gcd(𝑚1, 𝑚2) (Euclid’s alg) 2. Check 𝑑 = 1 (this means there is a unique sol mod 𝑚1𝑚2 3. Write 1 as a linear comb of 𝑚1 and 𝑚2, 1 = 𝑠1𝑚1 + 𝑠2𝑚2 4. Now 𝑥 ≡ 𝑐1𝑚2𝑠2 + 𝑐2𝑚1𝑠1(𝑚𝑜𝑑 𝑚1𝑚2) ALT: multiply each c by the product of all other moduli, and sum ALT: solve one and write in terms of 𝑎 + 𝑏𝑛 ∀𝑛 ∈ ℤ, them substitute into second and solve for n 3 Polynomials Monic – leading coeff is 1 Degree 0: non-zero constant Degree −∞: zero constant “the zero polynomial” Binomial Theorem (𝑥 + 𝑦)𝑛 = ∑ (𝑛 𝑗 )𝑛 𝑗=0 𝑥𝑗𝑦𝑛−𝑗 where (𝑛 𝑗 ) = 𝑛! 𝑗!(𝑛−𝑗)! Results: set x = y = 1 for 2𝑛 = ∑ (𝑛 𝑗 )𝑛 𝑗=0 Can also set x = 1 and y = -1, or differentiate wrt x and set x = y = 1: Remainder Theorem For any polynomials p and q, where n and m are the degrees, there is a polynomial s of degree 𝑚 – 𝑛 and a polynomial m of degree < 𝑚 s.t. 𝑝(𝑥) = 𝑞(𝑥)𝑠(𝑥) + 𝑟(𝑥) ∀𝑥 – use polyn long division to find If 𝑟(𝑥) = 0 ∀𝑥 then q is a factor of p Euclid’s Algorithm for Polynomials If q a factor of p, then so is 𝑐𝑞 Highest common factor must be monic Common factors of p and q are also factors of r, so Euclid’s algorithm can be used in the same way as for integers We can also write the hcf as a linear combination Roots of Polynomials Roots of monic polynomials with integer coeffs: algebraic integers Roots of polynomials with integer coeffs: algebraic numbers Otherwise transcendental (e.g. 𝜋) Complex polynomials of degree 𝑛 have 𝑛 roots (Fundamental Theorem of Algebra), real polynomials have at most 𝑛 Any real polynomial can be factored into linear and quadratic terms 4 Vectors Right hand rule for 𝒊, 𝒋, 𝒌, also 𝒊 × 𝒋 = 𝒌 Dot prod: 𝒖 ⋅ 𝐯 = |𝒖||𝒗| cos 𝜃 “how much of u in direction of v?”. For unit vectors ranges between -1 and 1. Commutative, 𝒖 ⋅ 𝒖 = |𝒖|2, distributive over vector addition Cross prod: |𝒖 × 𝒗| = |𝒖||𝒗| sin 𝜃, direction is perp to both u and v by the right hand rule. Anti-commutative (direction reverses), 𝒖 × 𝒖 = 𝟎, distributive over vector addition. To calculate, put into a 3x3 matrix with 𝒊, 𝒋, 𝒌 on top row. Magnitude is also area of parallelogram Vector equation of a line: 𝒙 = 𝒑 + 𝑡(𝒒 − 𝒑) = 𝑡𝒒 + (1 − 𝑡)𝒑 Distance from point to line: find general vec from point to line, which must be perpendicular to the line Distance from point to plane: if 𝒙 ⋅ 𝒏 = 𝒑 ⋅ 𝒏 and the point is 𝒂, then distance is |𝒑⋅𝒏−𝒂⋅𝒏| |𝒏| 5 Matrices Row first, column second: General solution of a system of eqns: use a 𝜆 parameter if needed Echelon form: all entries one side of diagonal are zero (or better) and all zero rows are at the bottom Reduced echelon form: zeros above and below pivots Strict reduced echelon form: pivots are all 1 (easy to read off sols from augmented matrix) Rank = number of pivots Row ops: multiply (multiplies determinant), add to each other, interchange rows (negates determinant) For 𝐴𝒙 = 𝒃 where 𝐴 ∈ ℝ𝑚×𝑛, 𝒙 ∈ ℝ𝑛, 𝒃 ∈ ℝ𝑚, there is a sol for all b if rank(A) = m (cols = rank). Unique if also rank(A) = n (rows=rank). To find the inverse: • do row ops on the matrix augmented with the identity to get the identity on the left • Or use 𝐴−1 = 1 𝑑𝑒𝑡𝐴 ( 𝑑 −𝑏 −𝑐 𝑎 ) for a 2x2 matrix • For bigger, take 1 𝑑𝑒𝑡𝐴 𝐶𝑇 where C is the matrix of cofactors Determinant Only on square matrices 𝑑𝑒𝑡(𝐴𝐵) = 𝑑𝑒𝑡 𝐴 ⋅ 𝑑𝑒𝑡 𝐵 and 𝑑𝑒𝑡 𝐼 = 1