Explanation:

Definition of One-to-One Linear Transformation:

A linear transformation T from R^n to R^n is said to be one-to-one (injective) if each distinct vector in the domain R^n is mapped to a distinct vector in the codomain R^n.

Explanation of a Linear Transformation that is Not One-to-One:

When a linear transformation T from R^n to R^n is not one-to-one, it means that there exist distinct vectors in the domain R^n that are mapped to the same vector in the codomain R^n. This can happen for various reasons, such as:
- The transformation T collapses multiple vectors in the domain onto a single vector in the codomain.
- The transformation T projects vectors onto a lower-dimensional subspace, causing multiple vectors to be mapped to the same vector.
- The transformation T introduces a non-trivial kernel (null space), leading to non-invertibility and hence non-one-to-one mapping.

Example:

Consider a linear transformation T: R^2 -> R^2 defined by T(x, y) = (x, 0). This transformation maps all vectors in R^2 to the x-axis in R^2. Since all vectors of the form (x, 0) are mapped to the same vector (x, 0) in the codomain, this transformation is not one-to-one.

Conclusion:

In conclusion, a linear transformation from R^n to R^n that is not one-to-one implies that there exist distinct vectors in the domain that are mapped to the same vector in the codomain. This can occur due to collapsing multiple vectors, projection onto a lower-dimensional subspace, or the presence of a non-trivial kernel.