Parts

• Fundamentals
• Structures
• Summary

Linear map is a special type of homomorphism between vector spaces, which preserves the operations of vector addition and scalar multiplication : $$ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}), \quad T(c\mathbf{v}) = c T(\mathbf{v}) $$ So we can also treat linear map as a kind of transformation (or action) which is performed on vectors in the domain, resulting in vectors in the codomain.

Homomorphism

When talk about homomorphism, a few properties need to be considered, such as:

Invariant Subspaces

Definition 5.1 – Invariant Subspace

Let $T \in \mathcal{L}(V)$ be a (linear) operator on vector space $V$. When we restrict $T$ to a subspace $U \subseteq V$, we are not sure the corresponding vectors will still lie in $U$. If they do, we say that $U$ is an invariant subspace under $T$. We call $T|_U$ is an operator on $U$.

Eigenvalues and Eigenvectors

Inner Product Spaces