How can linear transformations be visually interpreted?

Linear transformations can be visually interpreted as the rotation and translation of geometric shapes. This stems from the fundamental properties of linear transformations, which preserve the operations of vector addition and scalar multiplication. When you apply a linear transformation to a geometric shape represented by vectors, the shape can be altered in various ways, including being rotated, reflected, or translated without changing its inherent characteristics, such as collinearity and ratios of distances.

For instance, if you consider a 2D vector space, applying a linear transformation can rotate the vectors defining a shape by a certain angle or translate them to a different position in the plane. These transformations maintain the parallelism of lines and the invariance of the origin point, reflecting the nature of linear maps in vector spaces.

Other options do not correctly convey the essence of linear transformations. Scaling of non-linear functions suggests a manipulation that does not fit the definition of linearity, as linear transformations specifically apply to linear functions and maintain proportionality and linearity. The reflection of scalar values over an axis implies a more limited operation than what linear transformations can represent as a whole. Lastly, the expansion of matrix spaces into polynomials does not describe the action of linear transformations, which typically involves operations on vectors in vector spaces rather than implying a