Category Archives: Linear Algebra

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… Continue reading →

Parts Fundamentals Structures Summary Continue reading →

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)… Continue reading →

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… Continue reading →

Table of Contents Table of Contents Polynomials Eigenvalues and Eigenvectors Inner Product Spaces Eigenvalues and Eigenvectors Inner Product Spaces

Subspace Definition 1.0 – Subspace of Vector Space A subspace $U$ of a vector space $V$ is a subset of $V$ that is itself a vector space under the same addition and scalar multiplication as $V$. Formally, it has two properties: Continue reading →