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

Dot products and duality

This quiz connects coordinate arithmetic with Euclidean geometry and scalar-valued linear maps. It accompanies Chapter 9 of 3Blue1Brown’s Essence of Linear Algebra.

Review vector coordinates, lengths, angles, projections and matrix-vector multiplication. Questions ask you to calculate dot products, interpret their signs, explain symmetry and scaling, and construct row matrices from basis images.

• Distinguish signed scalar projection from an unsigned length or vector projection.
• Relate a Euclidean vector to the linear functional defined by dotting with it.
• Check how unit and non-unit vectors affect projection interpretation.
• Identify violations of linearity and handle zero-vector or collapsed-direction cases.

The vector–functional correspondence here uses the standard real Euclidean dot product. Signed projections preserve directional information, and projection formulas involving a direction require a nonzero direction vector. Read multiple-select questions carefully and judge each statement independently.