Explore cubic splines for interpolation in this 42-minute lecture on linear algebra. Learn how to create smooth curves passing through a finite number of points in the (x,y) plane using Taylor coefficient vectors. Discover a novel approach that builds cubic splines segment by segment, transferring Taylor coefficient vector information between endpoints. Gain insights into the relationships between Taylor vectors using linear transformations and Pascal matrices. Develop calculus concepts independently, making the content accessible even without prior calculus knowledge. Enhance your understanding of polynomial interpolation and its applications in design and mathematics.