# Lecture - Linear Algebra MT22, XIII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/13/ · Updated: 2022-12-06 · Tags: uni, lecture

### Flashcards
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?::
1. Bilinear
2. Symmetric
3. Positive definite

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 $||v||$?::
$$
||v|| = \sqrt{\langle v, v \rangle} 
$$

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.

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