3Blue1Brown: Linear Algebra: Dot Products and Duality - Chapter 9 Quiz

This quiz rigorously tests your understanding of dot products, their geometric and algebraic interpretations, and their profound connection to linear transformations from a multi-dimensional space to the one-dimensional number line, as explained in 3Blue1Brown's 'Essence of Linear Algebra' Chapter 9: 'Dot products and duality'. The questions delve into the concept of duality, where vectors can be seen as the 'embodiment' of such linear transformations. Mastery of these concepts requires a nuanced understanding of:

Numerical Definition of Dot Product: For two vectors $v = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$ and $w = \begin{pmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{pmatrix}$, their dot product is given by:

$\quad v \cdot w = v_1 w_1 + v_2 w_2 + \dots + v_n w_n$

Geometric Interpretation of Dot Product: The dot product $v \cdot w$ can be interpreted as the length of the projection of $w$ onto $v$, multiplied by the length of $v$. The sign indicates direction:

$\quad v \cdot w = ||v|| \cdot ||w|| \cdot \cos(\theta)$, where $\theta$ is the angle between $v$ and $w$.

Linear Transformations to the Number Line: A linear transformation $T: \mathbb{R}^n \to \mathbb{R}$ maps vectors to scalars. It can be represented by a $1 \times n$ matrix $A = \begin{pmatrix} a_1 & a_2 & \dots & a_n \end{pmatrix}$. Applying this transformation to a vector $x$ results in:

$\quad T(x) = A x = \begin{pmatrix} a_1 & a_2 & \dots & a_n \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = a_1 x_1 + a_2 x_2 + \dots + a_n x_n$

Duality: The central idea that any linear transformation from a vector space to the number line corresponds to a unique vector in that space, such that applying the transformation is equivalent to taking the dot product with that vector. Conversely, any vector defines such a linear transformation. This quiz will challenge your conceptual grasp of these interconnections.