3Blue1Brown: Linear Algebra Chapter 13: Change of Basis and Coordinate Systems

This quiz assesses your understanding of coordinate systems, basis vectors, and the concept of changing basis in linear algebra. It covers how vectors and transformations are represented in different coordinate systems and the mathematical procedures required to translate between them. You will be tested on the role of basis vectors like i-hat and j-hat, the construction and application of change-of-basis matrices, and the interpretation of matrix compositions for transforming vectors and entire transformations from one 'language' or perspective to another.

Key Formulae:

1. Vector in a non-standard basis: A vector with coordinates $[c_1, c_2]$ in a system with basis vectors $\vec{b_1}$ and $\vec{b_2}$ is given by $c_1\vec{b_1} + c_2\vec{b_2}$.

2. Change of Basis (New to Standard): Let $A$ be the matrix whose columns are the new basis vectors (e.g., $\vec{b_1}, \vec{b_2}$) written in standard coordinates. To convert a vector $\vec{v}_{new}$ from the new system to the standard system ($\vec{v}_{std}$), you compute: $\vec{v}_{std} = A \cdot \vec{v}_{new}$.

3. Change of Basis (Standard to New): To convert a vector $\vec{v}_{std}$ from the standard system to the new system ($\vec{v}_{new}$), you compute: $\vec{v}_{new} = A^{-1} \cdot \vec{v}_{std}$.

4. Transforming a Transformation: A transformation represented by matrix $M$ in the standard system is represented by the matrix $M'$ in the new system, where $M' = A^{-1}MA$.