Linear transformations in two dimensions

• Linear transformations in two dimensions primarily refer to the mathematical operations conducted on geometric shapes on a plane.
• A linear transformation, also known as a linear map of the plane, is a particular type of transformation or function which is marked by its ability to preserve straight lines and maintain ratios of distances. This means that any straight line remains a straight line, and distances are enlarged or reduced uniformly.
• Understand that linear transformations operate through two main types: translations and scalings. A translation shifts a shape a certain distance in a particular direction without altering its size or orientation, whereas a scaling multiplies all spatial coordinates by a constant factor.
• Be familiar with vectors as they play a significant role in linear transformations. A vector in two dimensions is a directed line segment represented by coordinates (x, y). When considering linear transformations, vectors can represent any point on the plane.
• The process of linear transformation can use matrix multiplication. You can express linear transformations as 2x2 matrices where each column of the matrix represents the images of the unit vectors along the x and y axes.
• Comprehend that every 2x2 matrix represents a unique linear transformation, and each linear transformation has a unique matrix. This forms the basis of equivalence between matrices and linear transformations.
• Understand the concept of the determinant in a 2x2 matrix. The determinant has a crucial geometric interpretation: it gives you the area scaling factor by which the linear transformation multiplies areas. A determinant of zero implies that the transformation squishes the plane down to a line or a point.
• Remember that a transformation is invertible if and only if its matrix determinant is non-zero. The inverse transformation corresponds to the inverse of the matrix.
• Grasp the effects of combining transformations: the combination of two transformations is the matrix product of their matrices. The order of transformations is critical - matrix multiplication is not commutative.