autre said:If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.
micromass said:This is not true. Injectivity is not enough. You need bijectivity.
autre said:Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.
mtayab1994 said:but doesn't he have to use injectivity and surjectivity to show bijectivity?
A linear transformation is a mathematical operation that maps a vector from one vector space to another, while preserving the structure of the vector space. In simpler terms, it is a function that takes in a vector as input and outputs a different vector.
The inverse of a linear transformation is a function that "undoes" the original transformation, such that when the two functions are composed together, they result in the identity function. In other words, the inverse of a linear transformation "reverses" the effects of the original transformation.
A linear transformation can be represented by a square matrix. Each column of the matrix represents the transformation of a standard basis vector in the original vector space. The resulting vector after the transformation can be found by multiplying the matrix by the input vector.
Inverse linear transformations are important because they allow us to "undo" a transformation and retrieve the original vector. They are also useful in solving systems of linear equations and performing other mathematical operations.
No, not all linear transformations are invertible. A linear transformation is invertible if and only if its associated matrix is nonsingular, meaning it has a nonzero determinant. If the matrix is singular, the transformation is not invertible and the inverse does not exist.