Inner Product Space Quiz

Which of the following is not a property of an inner product?

• Linearity in the first argument

• Symmetry: ⟨u, v⟩ = ⟨v, u⟩

• Positive definiteness

• ⟨au, v⟩ = a⟨u, v⟩ for all a ∈ ℝ and u, v ∈ V

If ⟨u, v⟩ = 0, what can be concluded?

• u = v

• u is orthogonal to v

• u is a scalar multiple of v

• v = 0

The norm induced by an inner product is defined as:

• ⟨v, v⟩

• √⟨v, v⟩

• ⟨v, u⟩

• |v|²

The Cauchy-Schwarz inequality states that:

• ⟨u, v⟩ ≤ ‖u‖‖v‖

• |⟨u, v⟩| ≤ ‖u‖‖v‖

• ⟨u, u⟩ = ‖u‖²

• ‖u + v‖ = ‖u‖ + ‖v‖

Which of the following norms is induced by an inner product on ℝⁿ?

• ‖x‖₁

• ‖x‖₂

• ‖x‖∞

• None

In an inner product space, ⟨x, x⟩ = 0 implies:

• x is a unit vector

• x is orthogonal to itself

• x = 0

• None of the above

The triangle inequality states:

• ‖u + v‖ ≤ ‖u‖ + ‖v‖

• ⟨u, v⟩ ≤ ‖u‖ + ‖v‖

• ‖u − v‖ = ‖u‖ − ‖v‖

• ⟨u, u + v⟩ ≤ ‖u + v‖

In complex inner product spaces, ⟨u, v⟩ = _____________.

• ⟨v, u⟩

• ⟨v, u⟩̅

• ⟨u + v, u + v⟩

• None of the above

A set of vectors is orthonormal if:

• Each vector has norm 1

• Each pair is orthogonal

• Both A and B

• They span the space

The inner product ⟨f, g⟩ = ∫₀¹ f(x)g(x) dx defines an inner product on:

• ℝⁿ

• C[0, 1]

• ℝ

• None