Table of Contents
- 1. invertible, inverse def
- 1.1. careful
- 1.2. results
- 1.2.1. unique
- 1.2.2. equivalant to injectivity and surjectivity (bijectivity)
- 1.2.3. Equivalent Condition with eigenvalues
- 1.2.4. non-singular matrices are invertible
- 1.2.5. operators that are injective or surjective are bijective
- 1.2.6. matrices with linearly independent columns and rows are bijective
1 invertible, inverse def
- A linear map \(T \in \mathcal L(V, W)\) is invertible if there exists a linear map \(S\in \mathcal(W, V)\) such that \(ST\) equals the identity map on \(V\) and \(TS\) equals the identity map on \(W\).
- A linear map \(S \in \mathcal(W, V)\) satisfying \(ST = I\) and \(TS = I\) is called an inverse of \(T\)
- If \(T\) is invertable, \(T^{-1}\) denotes the inverse of \(T\)
1.1 careful
1.1.1 the inverse of a map has to be commutative (\(TS = I\) and \(ST = I\))
1.1.2 the target identity is in one space on one side and in the other space on the other side
1.2 results
1.2.1 unique
any invertible map has exactly one inverse
1.2.2 equivalant to injectivity and surjectivity (bijectivity)
See bijectivity. Iff a map is bijective, then it is invertable.
1.2.3 Equivalent Condition with eigenvalues
if a map has zero as an eigenvalue, then it is singular (5.A exercise 21)