Invariant subspace

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism. A subspace $W$ of $V$ is invariant under $T$, or $T$-invariant, when $T(w) \in W$ for all $w \in W$.

Remarks

The subspaces $0$ and $V$, $\ker T$ and $\im T$, and each eigenspace of $T$ are $T$-invariant. A line spanned by $v \ne 0$ is $T$-invariant if and only if $v$ is an eigenvector of $T$. If $V = W_1 \oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.

Explanations

L1 human

An invariant subspace is a part of the space that the map never leaves. Vectors inside it stay inside it. We can then study the map on this smaller space alone.