3Blue1Brown: Linear Algebra : Nonsquare Matrix Transformations between dimensions -Chapter 8 Quiz

Nonsquare matrices and transformations between dimensions

This quiz accompanies Chapter 8 of 3Blue1Brown’s Essence of Linear Algebra. It explores how rows and columns identify the codomain and domain of a matrix transformation, and how transformed input basis vectors determine its columns.

Prerequisites include vectors, linear combinations, spans and basic matrix multiplication. Questions cover concrete matrix construction, rank and nullity, composition, scalar-valued maps and their connection to dot products.

• Distinguish the codomain from the actual image or column space.
• Explain why mapping from higher to lower dimensions cannot be injective.
• Recognize when a map into a larger space is an embedding.
• Interpret full rank for tall and wide matrices.
• Check which matrix operations are defined for a given shape.

Use geometric intuition alongside the algebraic definition of linearity. A change in domain and codomain dimensions does not change those axioms, and a larger codomain does not guarantee a larger image. Both single-choice and multiple-select questions are included.