Final answer:
A linear transformation is a function that preserves vector addition and scalar multiplication. These transformations are not limited to three dimensions and do not inherently involve square roots or result in a straight line. The correct condition for a linear transformation is option a: It preserves addition and scalar multiplication.
Step-by-step explanation:
Understanding Linear Transformations
A linear transformation in mathematics is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The essence of a linear transformation is well-captured in its two main properties:
- Addition: If u and v are vectors in the vector space, the linear transformation T satisfies T(u + v) = T(u) + T(v).
- Scalar Multiplication: For a scalar c and a vector u in the vector space, the linear transformation T satisfies T(cu) = cT(u).
These properties ensure that linear transformations are dimensionally consistent, meaning they can work in any number of dimensions, not only in three dimensions. Moreover, linear transformations do not inherently involve square roots; rather, their primary operation lies in linear equations and preserving vector operations. The representation of a linear equation, typically y = mx + b, demonstrates the straight-line graph that results from such an equation. Nonetheless, the fact that the graph of a linear equation is a straight line should not be conflated with the misconception that a linear transformation's output is always a straight line — the output is determined by the dimensions of the input and target spaces.
Considering the given options and the properties of linear transformations, it is clear that option a is correct: A linear transformation preserves addition and scalar multiplication.