# Lecture - Linear Algebra MT22, IV

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/4/ · Updated: 2022-10-28 · Tags: uni, lecture

What is the condition for $A$ to be an orthogonal matrix?::
$$
A^\top = A^{-1}
$$

What is $(Ax)^\top (Ay)$ when $A$ is an orthogonal matrix?::
$$
x^\top y
$$

What is special about $Ax$ when $A$ is an orthogonal matrix?::
It doesn’t change the length of the vector.

What are the two operations on a vector space?::
- Vector addition
- Scalar multiplication

Other than vector addition and scalar multiplication forming a group, what is the extra condition about scalar multiplication?::
It is distributive.

What name is given to the scalars in a vector space?::
The base field (typically $\mathbb{R}$).

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