Lecture - Linear Algebra MT22, XIII

If $X$ is a change of basis matrix, and $G$ is a Gram matrix, what is the Gram matrix $H$ with respect to the new basis?

\[H = X^\intercal G X\]

What are the three conditions for a bilinear form $B$ to be an inner product?

What does it mean for a bilinear form $B$ to be positive definite?

\[\forall v \in V, \text{ }B(v, v) \ge 0 \text{ and } B(v,v) = 0 \text{ iff } v = 0\]

What do you call a real vector space equipped with an inner product?

An inner product space.

If $H$ is a symmetric, positive definite matrix, how can you construct an inner product $\langle\cdot, \cdot\rangle$

\[\langle v, w\rangle = x^\intercal H y\]

where $x$ and $y$ are coordinate representations.

What is the name for $\mathbb{R}^n$ equipped with the dot product as an inner product called?:: $n$-dimensional Euclidean space.

In terms of an inner product $\langle v, w \rangle$, what is the formula for the norm or ‘length’ of $ \vert \vert v \vert \vert $?

\[||v|| = \sqrt{\langle v, v \rangle}\]

In terms of an inner product $\langle v, w \rangle$, what is the formula for the ‘angle’ $\theta$ between $v$ and $w$?

\[\theta = \cos^{-1}\frac{\langle v, w\rangle}{||v||\text{ }||w||}\]

If the angle between two vectors (when using an inner product) is $\frac{\pi}{2}$, what do you say about those two vectors?

They are orthogonal.