Final answer:
Linear transformations preserve the properties of lines or linear relationships. They are represented by matrices and have important properties such as being closed under addition and scalar multiplication.
Step-by-step explanation:
Linear transformations are mathematical functions that preserve the properties of lines or linear relationships. They are represented by matrices and can be used to describe the motion of objects in physics, the correlation between variables in statistics, or the mapping of points in geometry. In linear transformations, lines are transformed into lines, parallel lines remain parallel, and the origin remains fixed.
For example, consider the transformation T(x) = 2x. This transformation scales every point on a line by a factor of 2. If the original line passes through the point (1,1), the transformed line will pass through the point (2,2), maintaining the same slope.
Linear transformations have many important properties, such as being closed under addition and scalar multiplication, preserving linear combinations, and having an inverse for invertible transformations.