A linear transformation one-to-one is a function between two vector spaces that preserves the algebraic operations of addition and scalar multiplication. This means that for every input, there is a unique output, and no two inputs will result in the same output. In other words, the function maps each element of the domain to a different element in the range.
To determine if a linear transformation is one-to-one, we can use the nullity-rank theorem. This theorem states that the nullity of a linear transformation is equal to the dimension of the null space, and the rank of the linear transformation is equal to the dimension of the range. If the nullity and rank are equal, then the linear transformation is one-to-one.
A one-to-one linear transformation maps each element of the domain to a different element in the range, while an onto linear transformation maps every element in the range to at least one element in the domain. In other words, a one-to-one linear transformation has a unique output for each input, while an onto linear transformation has a full range of outputs.
Yes, a linear transformation can be both one-to-one and onto. This type of linear transformation is called an isomorphism, which means it preserves both the algebraic operations and the structure of the vector spaces. In other words, an isomorphism is a bijective linear transformation.
Some real-life examples of one-to-one linear transformations include scaling and rotation transformations in geometry, conversion between different units of measurement, and encryption and decryption functions in computer science. In each of these examples, there is a unique output for every input, and no two inputs will result in the same output.