Final answer:
A linear transformation is one-to-one if it does not map any different vectors to the same vector. A linear transformation maps onto if every vector in the codomain has at least one preimage in the domain.
Step-by-step explanation:
A linear transformation is one-to-one if and only if the kernel (null space) of the transformation is the zero vector. In other words, a linear transformation is one-to-one if and only if it does not map any two different vectors to the same vector.
A linear transformation maps onto if and only if the range (image) of the transformation is the entire codomain. In other words, a linear transformation maps onto if and only if every vector in the codomain has at least one preimage in the domain.
To determine whether the linear transformation t is one-to-one, we need to check whether the rank of the matrix a is equal to the number of columns in a. If the rank is equal to the number of columns, then the transformation is one-to-one. To determine whether the linear transformation t maps onto, we need to check whether the rank of the matrix a is equal to the number of rows in a. If the rank is equal to the number of rows, then the transformation maps onto.