Summary of Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra

00:00:00 - 00:10:00

Cramer's rule is a way of finding the coordinates of an unknown input vector in a linear system of equations. The rule works by taking the determinant of the transformation matrix and solving for the desired coordinate. This method is helpful in cases where the linear system is too complex to solve by traditional means.

• 00:00:00 In this video, Cramer's rule, explained geometrically, is discussed. The relevant background needed here is an understanding of determinants, dot products, and linear systems of equations. Cramer's rule is not the best way for computing solutions to linear systems of equations, but it is helpful to see how these concepts relate to each other. From a purely artistic standpoint, the result of applying Cramer's rule to a linear system is often very pleasing.
• 00:05:00 Cramer's rule states that the coordinates of a vector after a matrix transformation will be the same as the coordinates of the vector before the matrix transformation, provided that the determinant of the transformation matrix is the same. This is useful in recovering the y-coordinate of a mystery input vector in a linear system of equations.
• 00:10:00 Cramer's rule tells us how to find the x- and y-coordinates of an unknown input vector in a linear system of equations. The rule works for three dimensions as well as for two dimensions, and can be generalized to higher dimensions if needed. Learning how to apply Cramer's rule is key to solving linear systems.