- #1
mbud
- 7
- 0
I was just wondering how you know if linear transformations injective?
An injective linear transformation is a function that preserves vector addition and scalar multiplication, and also maps each input vector to a unique output vector. This means that no two input vectors will have the same output vector, making it a one-to-one mapping.
An injective linear transformation is one-to-one, meaning that each input vector has a unique output vector. On the other hand, a surjective linear transformation is onto, meaning that every output vector has at least one corresponding input vector. In other words, an injective transformation does not have any "lost" output vectors, while a surjective transformation does not have any "extra" output vectors.
To determine if a linear transformation is injective, you can use the rank-nullity theorem. If the nullity (dimension of the null space) is 0, then the transformation is injective. Another way is to check if the determinant of the transformation's matrix is non-zero. If it is, then the transformation is injective.
Yes, an injective linear transformation can have a non-square matrix. The matrix will have more columns than rows, and it is still possible for the transformation to be injective as long as the determinant is non-zero.
An injective linear transformation is useful in applications where we want to map unique inputs to unique outputs, such as in cryptography or data compression. It also helps to reduce data redundancy and improve efficiency in data processing. In addition, injective transformations are used in linear algebra for solving systems of equations and finding inverse transformations.