# Lecture - Linear Algebra I MT22, XIV

### Flashcards

What is the Cauchy-Schwartz inequality relating $v$ and $w$ in an inner product space?

What is $ \vert \vert \alpha \underline{v} \vert \vert $ equal to?

How does the triangle inequality generalise for a norm $ \vert \vert u + v \vert \vert $?

What is the statement of the inequality of arithmetic and geometric means for $x _ 1, x _ 2, \ldots, x _ n$?

When does equality hold for the arithmetic and geometric means

\[\frac{x_1 + x_2 + \ldots + x_n}{n} \ge \sqrt[n]{x_1\cdot x_2 \cdot\ldots\cdot x_n}\]
?

When $x _ 1 = x _ 2 = \ldots = x _ n$.

(Best approximation in inner product spaces) If $V$ is an inner product space and $W$ is a subspace of $V$, if for any $v \in V$ then $w\in W$ satifies $\langle v - w, r \rangle = 0 \forall r \in W$ then what must be true about the distance from $v$ to $w$ and from $v$ to any other vector $u \in W$?

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about linearity?

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about symmetry?

Why don’t the axioms for an inner product space over $\mathbb{C}$ specify that the inner product is bilinear?

Because the symmetric conjugate rule means that it isn’t.

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about positive definiteness?

Why is $\langle v, v \rangle$ guarenteed to be real in an complex inner product space?:: Because the symmetric-conjugate condition.