Lecture - Linear Algebra I MT22, XIV

Created: December 06, 2022

Flashcards

What is the Cauchy-Schwartz inequality relating $v$ and $w$ in an inner product space?
\[|\langle v, w \rangle| \le ||v|| \text{ } ||w||\]

What is $ \vert \vert \alpha \underline{v} \vert \vert $ equal to?
\[|\alpha|\text{ }||v||\]

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

What is the statement of the inequality of arithmetic and geometric means for $x _ 1, x _ 2, \ldots, x _ n$?
\[\frac{x_1 + x_2 + \ldots + x_n}{n} \ge \sqrt[n]{x_1\cdot x_2 \cdot\ldots\cdot x_n}\]

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 $u \in W$?
\[||v - w|| \le ||v - u||\]

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about linearity?
\[\langle \alpha u + \beta v, w\rangle = \alpha\langle u, w\rangle + \beta \langle v, w \rangle\]

In a vector space over $\mathbb{C}$, the axioms for an inner product space are different. What is the axiom about symmetry?
\[\langle u, v \rangle = \overline{\langle v, u \rangle}\]

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?
\[\langle v, v \rangle \ge 0 \text{ with } v = 0 \iff \langle v, v \rangle = 0\]

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

What is the conjugate transpose, $A^*$?
\[A^* = \overline{A^\intercal}\]

What does it mean for a matrix $A$ to be unitary?
\[A^\*A = AA^\* = I\]

What’s the analogue of the dot product $x \cdot y = x^\intercal y$ for a complex vector space?
\[x^* y\]


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