3Blue1Brown: 3D Transformations - Chapter 5 Quiz

This quiz delves into the fundamental concepts of three-dimensional linear transformations as presented in the 3Blue1Brown "Essence of Linear Algebra" series, specifically Chapter 5. It tests your understanding of how 2D linear algebra principles extend to 3D space, the role of basis vectors in defining transformations, matrix representation of these transformations, and the mechanics of vector and matrix multiplication in three dimensions. Expect questions that require critical thinking, application of theoretical knowledge, and a strong grasp of vector and matrix operations.

Important Formulae:

• Standard basis vectors in 3D:
  • $\hat{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$
  • $\hat{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$
  • $\hat{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$
• Representation of a vector $\vec{v}$ in terms of basis vectors:
  • $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = x\hat{i} + y\hat{j} + z\hat{k}$
• Matrix representation of a 3D linear transformation $T$:
  • If $T(\hat{i}) = \begin{pmatrix} a_{11} \\ a_{21} \\ a_{31} \end{pmatrix}$, $T(\hat{j}) = \begin{pmatrix} a_{12} \\ a_{22} \\ a_{32} \end{pmatrix}$, and $T(\hat{k}) = \begin{pmatrix} a_{13} \\ a_{23} \\ a_{33} \end{pmatrix}$, then the transformation matrix $A$ is: $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$
• Applying a transformation $T$ to a vector $\vec{v}$ (Matrix-Vector Multiplication):
  • $T(\vec{v}) = A\vec{v}$
  • $A\vec{v} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = x \begin{pmatrix} a_{11} \\ a_{21} \\ a_{31} \end{pmatrix} + y \begin{pmatrix} a_{12} \\ a_{22} \\ a_{32} \end{pmatrix} + z \begin{pmatrix} a_{13} \\ a_{23} \\ a_{33} \end{pmatrix} = \begin{pmatrix} a_{11}x + a_{12}y + a_{13}z \\ a_{21}x + a_{22}y + a_{23}z \\ a_{31}x + a_{32}y + a_{33}z \end{pmatrix}$
• Composition of transformations (Matrix-Matrix Multiplication):
  • If $T_1$ is represented by matrix $A$ and $T_2$ by matrix $B$, then applying $T_2$ first, then $T_1$, is represented by the matrix product $AB$.
  • For $3 \times 3$ matrices $A = [a_{ij}]$ and $B = [b_{ij}]$, the element $(AB)_{ij}$ is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$.
  • $(AB)_{ij} = \sum_{k=1}^3 a_{ik}b_{kj}$