3Blue1Brown: Linear Transformations and Matrices - Chapter 3 Quiz

This quiz rigorously tests your understanding of 2D linear transformations and their relationship to matrices, based on the foundational concepts presented in Chapter 3 of 3Blue1Brown's 'Essence of linear algebra' series. It covers visual interpretations of linearity, the role of basis vectors, matrix construction, matrix-vector multiplication as linear combinations, and the implications of various transformation types. Questions are designed to be challenging, requiring a deep conceptual and computational understanding of the topic.

Key Formulas and Concepts:

• Definition of Linear Transformation (Visual):

  1. All lines must remain lines without getting curved.
  2. The origin must remain fixed in place. (Implies grid lines remain parallel and evenly spaced.)

• Definition of Linear Transformation (Algebraic): A transformation $L$ is linear if for all vectors $\vec{v}, \vec{w}$ and scalar $c$:

  1. Additivity: $L(\vec{v} + \vec{w}) = L(\vec{v}) + L(\vec{w})$
  2. Homogeneity/Scaling: $L(c\vec{v}) = cL(\vec{v})$ (A consequence of these properties is $L(\vec{0}) = \vec{0}$)

• Representing 2D Linear Transformations with Matrices: A 2D linear transformation is completely described by where the standard basis vectors $\hat{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\hat{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ land. If $L(\hat{i}) = \begin{pmatrix} a \\ c \end{pmatrix}$ and $L(\hat{j}) = \begin{pmatrix} b \\ d \end{pmatrix}$, then the transformation matrix $A$ is: $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

• Matrix-Vector Multiplication as Linear Combination: To find where a vector $\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ lands under the transformation represented by matrix $A$, we compute $A\vec{v}$: $A\vec{v} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = x \begin{pmatrix} a \\ c \end{pmatrix} + y \begin{pmatrix} b \\ d \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$ This represents the original vector's components ($x, y$) as scaling factors for the transformed basis vectors.

• Geometric Interpretations:

  • Rotation: e.g., 90° CCW is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
  • Shear: e.g., x-shear is $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$.
  • Scaling: e.g., uniform scaling by $k$ is $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$.
  • Reflection: e.g., across x-axis is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.
  • Linear Dependence of Columns: If $L(\hat{i})$ and $L(\hat{j})$ are linearly dependent, the transformation squishes 2D space onto a 1D line.