MATH 414 Lecture 17

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End Exam 1 content

Inner Products

Standard Inner products on $ℝ^{n}$ , $ℂ^{n}$ , $L^{2}$ , and $ℓ^{2}$
Is ⟨Y,X⟩=YTAX an inner product (given a matrix A)?
- Positivity: $⟨ \vec{v}, \vec{v} ⟩ > 0$ for all nonzero $\vec{v}$
- Homogeneity: $⟨ c \vec{u}, \vec{v} ⟩ = c ⟨ \vec{u}, \vec{v} ⟩$ for all vectors $\vec{u}, \vec{v}$ and scalars $c \in ℂ$ .
- (Conjugate) symmetry: $\overline{⟨ \vec{u}, \vec{v} ⟩} = ⟨ \vec{v}, \vec{u} ⟩$ for all vectors $\vec{u}, \vec{v}$
- Linearity: $⟨ \vec{u} + \vec{v}, \vec{w} ⟩ = ⟨ \vec{u}, \vec{w} ⟩ + ⟨ \vec{v}, \vec{w} ⟩$
Angle between vectors: $\cos θ = \frac{\vec{u} \cdot \vec{v}}{‖ \vec{u} ‖ ‖ \vec{v} ‖}$
Length of a vector: $‖ \vec{u} ‖ = \sqrt{⟨ \vec{u}, \vec{u} ⟩}$
Distance between two vectors: $d = ‖ \vec{u} - \vec{v} ‖$

If $f (x) = a_{0} + \sum_{k = 1}^{\infty} a_{k} \cos (k x) + b_{k} \sin (k x) \in L^{2} [- π, π]$ , then

\frac{1}{π} \int_{- π}^{π} {| f (x) |}^{2} d x = 2 {| a_{0} |}^{2} + \sum_{k = 1}^{\infty} {| a_{k} |}^{2} + {| b_{k} |}^{2}

If $f (x) = \sum_{k = - \infty}^{\infty} α_{k} e^{i k x} \in L^{2} [- π, π]$ , then

\frac{1}{2 π} {‖ f ‖}^{2} = \frac{1}{2 π} \int_{- π}^{π} {| f (x) |}^{2} d x = \sum_{k = - \infty}^{\infty} {| α_{k} |}^{2}

Moreover, for $f, g \in L^{2} [- π, π]$ , we get

\frac{1}{2 π} ⟨ f, g ⟩ = \frac{1}{2 π} \int_{- π}^{π} f (t) \overline{g (t)} d t = \sum_{n = 0}^{\infty} α_{n} \overline{β_{n}}