Vector
Calculus & Advanced MathA vector is a mathematical object with both magnitude (size) and direction, represented by an arrow or as a list of components.
Formula
v = \langle v_1, v_2, v_3 \rangle
Definition
A vector has two properties: size (magnitude) and direction. An arrow pointing northeast with a certain length is a vector. Just a length with no direction is a scalar.
Example
Wind blowing $20$ mph toward the north is a vector. Temperature of $70$ degrees is a scalar (just a number, no direction).
Key Insight
Vectors let mathematics describe things that point somewhere: velocity, force, displacement, electric fields.
Definition
A vector in $\mathbb{R}^n$ is an ordered $n$-tuple of real numbers $v = (v_1, v_2, \ldots, v_n)$. Vectors can be added component-wise: $u + v = (u_1+v_1, \ldots, u_n+v_n)$, and scaled: $cv = (cv_1, \ldots, cv_n)$. The magnitude (length) is $|v| = \sqrt{v_1^2 + \ldots + v_n^2}$.
Example
$v = (3, 4)$: $|v| = \sqrt{9+16} = \sqrt{25} = 5$. Adding $u = (1, 2)$ and $v = (3, 4)$: $u + v = (4, 6)$.
Key Insight
Vectors in physics combine naturally: displacement vectors add (walking 3 east then 4 north gives a net 5 units northeast), justifying the definition.
Definition
A vector space over field $F$ is a set $V$ with operations of addition and scalar multiplication satisfying $8$ axioms (closure, associativity, commutativity, identity, inverses, scalar distributivity). $\mathbb{R}^n$, polynomials, continuous functions, and matrices are all vector spaces. Linear maps between vector spaces are the morphisms of linear algebra.
Example
The space of continuous functions $C([0,1])$ is an infinite-dimensional vector space. The Fourier series represents functions as infinite linear combinations of basis vectors $\sin(nx)$ and $\cos(nx)$.
Key Insight
Abstraction to vector spaces unifies diverse mathematics: PDE solutions form function spaces, quantum states are vectors in Hilbert space, and error-correcting codes are subspaces of F_2^n.