3Blue1Brown: Linear Transformations and Matrices - Chapter 3 Quiz

Linear transformations and matrices

This quiz follows Chapter 3 of 3Blue1Brown’s Essence of Linear Algebra. It practices the connection between movement in the plane and calculations with matrices.

Practice topics

• Recognize linear maps and distinguish nonzero translations and nonlinear coordinate rules.
• Build a matrix from standard basis images.
• Compute matrix-vector products and recover input vectors from simple equations.
• Interpret rotations, reflections, shears, scalings and projections.
• Relate dependent columns to loss of dimension and invertibility.

A linear map satisfies $$L(u+v)=L(u)+L(v)$$ and $$L(cu)=cL(u)$$. With column-vector coordinates, $$\begin{pmatrix}a&b\c&d\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$$.

Prior familiarity with vectors, linear combinations and Cartesian coordinates is useful. Some questions require multiple selections. Check every alternative independently and calculate before reading the explanation. Remember that a singular linear map may collapse the plane to a line or a point; visual rules must allow these cases. The quiz introduces geometric interpretation rather than a complete treatment of determinants or eigenvalues.