Final answer:
True, a one-to-one mapping in linear algebra ensures unique vector mapping for each vector in the domain R^n to a unique vector in the codomain R^m. Additional concepts like vector addition being commutative and the difference between distance and displacement are also addressed.
Step-by-step explanation:
The statement, A mapping T: Rⁿ -> Rᵐ is one-to-one if each vector in Rⁿ maps onto a unique vector in Rᵐ, is True. A one-to-one mapping does ensure that each vector in the domain maps to a unique vector in the codomain, meaning no two different vectors in the domain map to the same vector in the codomain. This concept is fundamental in linear algebra when discussing functions and linear transformations. The term one-to-one specifically refers to the uniqueness of the mapping and does not depend on the matrices being square, nor is it inapplicable to linear transformations.
To provide additional information, it's important to note that a vector can indeed form the shape of a right angle triangle with its x and y components, reflecting the geometric interpretation of vectors in two-dimensional space. Furthermore, the addition of vectors is commutative, which implies that A + B = B + A, and this applies to any number of vectors in any dimension. Lastly, in one-dimensional motion, a scenario with zero distance and nonzero displacement cannot occur since distance is the magnitude of motion without direction, whereas displacement includes both magnitude and direction.