Consider a [direct sum of vector spaces,](Direct%20sums%20of%20vector%20spaces.md) [$\mathcal{V}=\mathcal{V}_1\oplus...\oplus \mathcal{V_n}$](Direct%20sums%20of%20vector%20spaces.md#^6f7b6b) where $\mathcal{V}_i$ is a [proper subspace.](Vector%20spaces.md#^e57c5a) For an [operator](linear%20operator) $A$ on $\mathcal{V},$ $\mathcal{V}_i$ is _invariant_ under $A$ if for some [vector,](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) $\mathbf{v}_i\in\mathcal{V}_i,$ $A\mathbf{v}_i\in\mathcal{V}_i.$ This means that $\mathcal{V}_i$ is invariant under $A$ if $A$ is an operator on $\mathcal{V}_i.$ # The trivial invariant subspace We consider the [zero vector space](Vector%20spaces.md#The%20zero%20vector%20space) to also be a _trivial invariant subspace_ since $\mathbf{0}\rightarrow \mathbf{0}$ is the only [linear map](Linear%20map.md) in $\{\mathbf{0}\}.$ %%Wikipedia also considers R^n (the n dimensional Euclidian space) as a trivial invariant subspace and I don't think so - or at least it isn't trivial in the way I use the term trivial.%% %%This entry on invariant subspacs is from page 76 of Axler's linear algebra%% # Invariant subspace problem ([... see more](Invariant%20subspace%20problem)) #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces