Wk 1: The vector

Question 1

ℝⁿ

Answer 1

Set of all n-vectors over ℝ

Answer 2

Set of functions from {0,1,2,…,d-1} to

Question 3

Sparse vector

Answer 3

a vector most of whose values are zero

Question 4

k-sparse vector

Answer 4

vector with no more than k nonzero entries

Question 5

lossy compression

Answer 5

represents a signal as sparse while preserving perceptual similarity

Question 6

Examples that can be represented by a vector

Answer 6

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

Question 7

Translation of Complex number

Answer 7

f(z) = z + z₁

z,z₁ ∈ ℂ

Question 8

Vector addition

Answer 8

[u₁, u₂, …, u_n] + [v₁, v₂, …, v_n] = [u₁ + v₁, u₂ + v₂, …, u_n + v_n]

Question 9

zero vector

Answer 9

the D-vector whose entries are all zero, written 0_D or just 0

Question 10

Basic properties of vector addition

Answer 10

associative and commutative

Question 11

scalar

Answer 11

term for field elements (used to scale vectors)

Question 12

Scalar-vector multiplication

Answer 12

α[v₁, v₂, …, v_n] = [αv₁, αv₂, …, αv_n]

Question 13

Basic property of scalar-vector multiplication

Answer 13

Associativity: α(βv) = (αβ)v

Question 14

{αv : α ∈ ℝ, 0 ≤ α ≤ 1}

Answer 14

line segment between the origin and v

Question 15

{αv : α ∈ ℝ}

Answer 15

line through the origin and v

Question 16

Distributive property of scalar-vector multiplication

Answer 16

α (u + v) = αu + αv

Question 17

Alt. representation of line segment

(sym. wrt end points)

Answer 17

{α[x¹₁, x¹₂] + β[x²₁, x²₂] : α,β ∈ ℝ, α,β ≥ 0, α + β = 1}

Question 18

convex combination of u and v

Answer 18

expression of the form αu + βv, where 0 ≤ α,β ≤ 1 and α + β = 1

Question 19

affine combination of u and v

Answer 19

an expression of the form αu + βv where α + β = 1

Question 20

Dot-product of two D-vectors

Answer 20

u ∙ v = ∑_k∈D u[k] v[k]

u ∙ v = u₁v₁ + u₂v₂ + … + u_nv_n

(aka scalar product)

Question 21

linear equation

Answer 21

equation of the form a ∙ x = β, where a is a vector, x is a vector of variables and β is a scalar

Question 22

Measuring similarity

Answer 22

Dot product can be used to measure similarity of two vectors (eg. voting records, audio samples)

Question 23

Algebraic properties of dot product

Answer 23

Commutativity: v ∙ x = x ∙ v

Homogeneity: αu ∙ v = α(u ∙ v)

Distributive law: (v₁ + v₂) ∙ x = v₁ ∙ x + v₂ ∙ x

Question 24

Solving a triangular system of equations

Answer 24

Use backward substitution