Linear Algebra Done Wrong

For these flashcards, I will be using Sergei Treil' Linear Algebra Done Wrong book

What is a vector space

A collection of vectors that follow the 8 axioms of vector spaces. These axioms are commutativity, associativity, the zero vector, additive inverse, multiplicative identity, multiplicative associativity and multiplying by scalars. Vector spaces are usually denoted by a V (1.1)

The space \mathbb{R}^n represents what

It represents a column vector of size n whose entries are real numbers (1.1)

The space \mathbb{C}^n represents what

It represents a column vector of size n whose entries are complex numbers (1.1)

The space \mathbb{F}^n represents what

It represents a column vector of size n whose entries are real or complex numbers (1.1)

An m x n matrix has how many rows and columns

This matrix has m rows and n columns. Matrices are always represented rows x columns (1.1)

The transpose of a matrix does what

Transforms the rows of a matrix into columns. So the first row becomes the first column and etc. The notation for the transpose of a matrix A is usually A^T (1.1)