MML — Analytic Geometry

Question 1

Inner product is

Answer 1

a positive definite, symmetric bilinear mapping Ω:V×V->ℝ. We typically write ⟨x, y⟩ instead of Ω(x, y)

Question 2

Euclidean vector space

Answer 2

is a vector space with inner product defined as dot product

Question 3

Symmetric, positive definite matrix

Answer 3

A symmetric real n×n matrix A that satisfies
∀x∈V{θ} : x^T A x > 0
is called symmetric, positive definite, or just positive definite. If only ≥ holds, then A is called symmetric, positive semidefinite.

Question 4

Cauchy-Schwarz Inequality

Answer 4

For an inner product vector space the introduced norm satisfies the Cauchy-Schwarz inequality
| ⟨x, y⟩ | ≤ ||x|| ||y||

Question 5

Metric

Answer 5

d(x, y) := ||x - y|| is called distance between x, y.
The mapping
d: V×V -> ℝ that is (x, y) -> d(x, y)
is called a metric.

Question 6

Definition of projection

Answer 6

Let V be a vector space and U⊆V a subspace of V. A linear mapping 𝜋: V->U is called a projection if 𝜋^2 = 𝜋∘𝜋 = 𝜋.