MATH 572 Section 6.A: Inner Products and Norms

Vocabulary practice flashcards covering definitions, properties, and fundamental theorems of inner products and norms from MATH 572 Section 6.A.

Dot Product

For $x, y \in \mathbf{R}^n$, the product defined by $x \cdot y = x_1y_1 + \dots + x_ny_n$, where $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$.

Inner Product

A function that assigns each ordered pair $(u, v)$ of vectors in $V$ a scalar $\langle u, v \rangle \in \mathbf{F}$ satisfying positivity, definiteness, additivity in the first slot, homogeneity in the first slot, and conjugate symmetry.

Positivity (Inner Product Property)

The condition that $\langle v, v \rangle \ge 0$ for all $v \in V$.

Definiteness (Inner Product Property)

The condition that $\langle v, v \rangle = 0$ if and only if $v = 0$.

Additivity in the First Slot

The inner product property stating that $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ for all $u, v, w \in V$.

Homogeneity in the First Slot

The inner product property stating that $\langle \lambda u, v \rangle = \lambda \langle u, v \rangle$ for all $\lambda \in \mathbf{F}$ and all $u, v \in V$.

Conjugate Symmetry

The inner product property stating that $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$.

Inner Product Space

A vector space $V$ together with an inner product on $V$.

Norm

For a vector $v \in V$, the real number defined by $\parallel v \parallel = \sqrt{\langle v, v \rangle}$.

Homogeneity of the Norm

The property that $\parallel \lambda v \parallel = |\lambda| \parallel v \parallel$ for every $\lambda \in \mathbf{F}$ and every $v \in V$.

Orthogonal Vectors

Two vectors $u, v \in V$ that satisfy $\langle u, v \rangle = 0$.

Orthogonality of the Zero Vector

The property stating that $0$ is orthogonal to every vector in $V$, and $0$ is the only vector in $V$ that is orthogonal to itself.

Pythagorean Theorem

The theorem stating that if $u, v \in V$ are orthogonal, then $\parallel u + v \parallel^2 = \parallel u \parallel^2 + \parallel v \parallel^2$.

Orthogonal Decomposition (6.14)

For $u, v \in V$ with $v \neq 0$, defining $c = \frac{\langle u, v \rangle}{\parallel v \parallel^2}$ and $w = u - \frac{\langle u, v \rangle}{\parallel v \parallel^2}v$ results in $\langle w, v \rangle = 0$ and $u = cv + w$.

Cauchy–Schwarz Inequality

The inequality stating that for all $u, v \in V$, $|\langle u, v \rangle| \le \parallel u \parallel \parallel v \parallel$, where equality holds if and only if one of $u, v$ is a scalar multiple of the other.

Triangle Inequality

The inequality stating that for all $u, v \in V$, $\parallel u + v \parallel \le \parallel u \parallel + \parallel v \parallel$, where equality holds if and only if one of $u, v$ is a nonnegative real multiple of the other.

Parallelogram Equality

The equality stating that for all $u, v \in V$, $\parallel u + v \parallel^2 + \parallel u - v \parallel^2 = 2(\parallel u \parallel^2 + \parallel v \parallel^2)$.