Wk 1: The vector Flashcards
ℝn
Set of all n-vectors over ℝ
Set of functions from {0,1,2,…,d-1} to
Sparse vector
a vector most of whose values are zero
k-sparse vector
vector with no more than k nonzero entries
lossy compression
represents a signal as sparse while preserving perceptual similarity
Examples that can be represented by a vector
document (for information retrieval)
binary string (for crypto / IT)
collection of attributes (voting record, demographic record, etc.)
state of a system
probability distribution
images
points
Translation of Complex number
f(z) = z + z1
z,z1 ∈ ℂ
Vector addition
[u1, u2, …, un] + [v1, v2, …, vn] = [u1 + v1, u2 + v2, …, un + vn]
zero vector
the D-vector whose entries are all zero, written 0D or just 0
Basic properties of vector addition
associative and commutative
scalar
term for field elements (used to scale vectors)
Scalar-vector multiplication
α[v1, v2, …, vn] = [αv1, αv2, …, αvn]
Basic property of scalar-vector multiplication
Associativity: α(βv) = (αβ)v
{αv : α ∈ ℝ, 0 ≤ α ≤ 1}
line segment between the origin and v
{αv : α ∈ ℝ}
line through the origin and v
Distributive property of scalar-vector multiplication
α (u + v) = αu + αv
Alt. representation of line segment
(sym. wrt end points)
{α[x11, x12] + β[x21, x22] : α,β ∈ ℝ, α,β ≥ 0, α + β = 1}
convex combination of u and v
expression of the form αu + βv, where 0 ≤ α,β ≤ 1 and α + β = 1
affine combination of u and v
an expression of the form αu + βv where α + β = 1
Dot-product of two D-vectors
u ∙ v = ∑k∈D u[k] v[k]
u ∙ v = u1v1 + u2v2 + … + unvn
(aka scalar product)
linear equation
equation of the form a ∙ x = β, where a is a vector, x is a vector of variables and β is a scalar
Measuring similarity
Dot product can be used to measure similarity of two vectors (eg. voting records, audio samples)
Algebraic properties of dot product
Commutativity: v ∙ x = x ∙ v
Homogeneity: αu ∙ v = α(u ∙ v)
Distributive law: (v1 + v2) ∙ x = v1 ∙ x + v2 ∙ x
Solving a triangular system of equations
Use backward substitution
