Three-dimensional linear transformations | Chapter 5, Essence of linear algebra

This video explains how linear transformations work in three-dimensional space, extending concepts from two dimensions. It demonstrates that any 3D linear transformation is fully determined by tracking where the standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land. The coordinates of these transformed basis vectors form the columns of a 3x3 matrix. A specific example shows a 90-degree rotation around the y-axis. The video further illustrates that multiplying a matrix by a vector corresponds to scaling and adding the transformed basis vectors according to the input vector's coordinates. Finally, it covers matrix multiplication as the composition of successive transformations.

Before you watch

• Two-dimensional linear transformations
• Basic properties of matrices and vectors
• Concept of basis vectors