Final answer:
A linear transformation matrix can be characterized as one-to-one if it maps each input to a unique output, onto if every output has at least one corresponding input, both if it possesses both of the properties, or neither if it has neither properties. This is determined by checking the matrix for full column rank (one-to-one) and full row rank (onto).
Step-by-step explanation:
The student has asked about the criteria for a linear transformation matrix being one-to-one or onto, or both. To determine this:
- One-to-one (injective) means that each element in the domain maps to a unique element in the codomain. In other words, no two different elements in the domain map to the same element in the codomain. A practical way to test this for a matrix transformation is to check whether the matrix has full column rank, meaning none of its columns are linearly dependent.
- Onto (surjective) means every element in the codomain has at least one element from the domain mapping to it. In matrix terms, this implies that the transformation covers the entire codomain space that is, it has full row rank.
- If a transformation is both one-to-one and onto, it is considered bijective and the matrix must have full rank, which is the minimum of the number of rows or columns.
- If it is neither one-to-one nor onto, then the matrix does not have full column or row rank, indicating that it is neither injective nor surjective.
Understanding the nature of a matrix in terms of these properties is crucial for many aspects of linear algebra, including solving systems of linear equations and understanding vector spaces.