Advanced Linear Algebra: Change of Basis

This quiz assesses your understanding of coordinate systems and the change of basis in linear algebra. You will be tested on your ability to translate vectors between different coordinate systems, understand the role of basis vectors, and determine how linear transformations are represented in alternate bases. The questions range from conceptual understanding to complex calculations, designed to challenge your grasp of these fundamental concepts.

Key Formulae:

Let the standard basis be $S = \{\hat{i}, \hat{j}\}$ and an alternate basis be $B = \{b_1, b_2\}$. The change of basis matrix $A$ from basis $B$ to $S$ is formed by using the basis vectors of $B$ as columns, expressed in standard coordinates: $A = [b_1 | b_2]$.

1. Translating a vector from basis B to the standard basis S:

   If a vector $v$ has coordinates $[x', y']^T$ in basis $B$ (denoted $v_B$), its coordinates in the standard basis $S$ (denoted $v_S$) are found by:

   $v_S = A \cdot v_B$

2. Translating a vector from the standard basis S to basis B:

   If a vector $v$ has coordinates $[x, y]^T$ in the standard basis $S$, its coordinates in basis $B$ are found by:

   $v_B = A^{-1} \cdot v_S$

3. Representing a linear transformation in an alternate basis:

   If a linear transformation is represented by matrix $M_S$ in the standard basis, its representation $M_B$ in basis $B$ is given by the similarity transformation:

   $M_B = A^{-1} \cdot M_S \cdot A$