Algebra
18 concepts
2x + 3Variables & Expressions
'Double some number and add 3' written compactly with the letter x as 2x+3.
ax + b = 0 → x = −b/aLinear Equations
An equality that's true for exactly one value of x. Finding that value is what 'solving the equation' means.
a₁x + b₁y = c₁, a₂x + b₂y = c₂Systems of Linear Equations
Finding the one pair (x, y) that satisfies both unknowns at once. You need two equations to pin the answer to a single point.
ax + b > 0Linear Inequalities
A statement that x is greater or less than something. The answer isn't one number but a 'range'.
y = ax + bLinear Functions & Their Graphs
A rule where you feed in x and out comes y = 'a times x plus b'. Graph it and you always get a straight line.
(a + b)² = a² + 2ab + b²Multiplication Formulas
(a+b)² is not a²+b²; a 2ab squeezes into the middle, making it a²+2ab+b².
x² + (a + b)x + ab = (x + a)(x + b)Factoring
Turning something like x²+5x+6 into a 'product of two expressions' such as (x+2)(x+3). It's the exact reverse of expanding.
ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / (2a)Quadratic Equations
An equation with an x² in it, ax²+bx+c=0. It has at most two solutions, and plugging a, b, c into the quadratic formula always produces them.
y = a(x − p)² + qQuadratic Functions
y = a(x−p)² + q. The graph is a parabola, and the vertex (p, q) is its 'pointed tip' (the highest or lowest point).
y = aˣ ⇔ x = logₐ yExponential & Logarithmic Functions
aˣ multiplies a by itself x times; logₐ y asks back 'how many times must I multiply a to reach y?'. Exponent and log undo each other.
aₙ = a₁ + (n−1)d, aₙ = a₁·rⁿ⁻¹Arithmetic & Geometric Sequences
An arithmetic sequence adds a fixed number d to the previous term each time; a geometric sequence multiplies the previous term by a fixed number r each time.
Σₖ₌₁ⁿ k = n(n+1)/2Sigma Notation & Series
Σ is a command: 'plug in from the bottom value up to the top value and add them all'. Σₖ₌₁ⁿ k is the sum from 1 to n.
(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼMatrices
A matrix arranges numbers in rows and columns. To multiply two matrices, you pair a 'row of the left' with a 'column of the right', multiply, and add.
(a + b)ⁿ = Σₖ₌₀ⁿ ₙCₖ aⁿ⁻ᵏ bᵏThe Binomial Theorem
When you expand (a+b)ⁿ, each term's coefficient is ₙCₖ — the number of ways to choose b from k of the n brackets.
T(v) = Av, T(u+v) = T(u)+T(v), T(cu) = c·T(u)Linear Transformations
A linear transformation moves every point of the plane (or space) while keeping grid lines straight, evenly spaced, and the origin fixed. This orderly motion is captured completely by a single matrix.
Av = λv (v ≠ 0); det(A − λI) = 0Eigenvalues & Eigenvectors
Apply the transformation A and most vectors get knocked off their direction. But certain special vectors v keep their direction and only get scaled by a factor λ. Such a v is an eigenvector, and the factor λ is its eigenvalue.
u, v ∈ V, a, b ∈ ℝ ⟹ a·u + b·v ∈ VVector Spaces
A vector space is a set whose elements never escape it when you add them together or multiply by a real number. Those two operations just need to obey a few natural rules.
det([[a, b], [c, d]]) = ad − bcThe Determinant
The determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. This number tells you the factor by which the matrix scales any shape's area.
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