I got this problem:
Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3
and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ respectively,
Find all the T invariant subspaces of $\mathbb{R}^3$
I found that:
All the T invariant subspace of dimension 0 are: $\{0\}$
All the T invariant subspace of dimension 1 are: $span\{v_1\}$ $span\{v_2\}$ and $span\{v_3\}$
All the T invariant subspace of dimension 3 are: $\mathbb{R}^3$
How do I show that all the T invariant subspace of dimension 2 are: $span\{v_1,v_2\}$ $span\{v_1,v_3\}$ and $span\{v_2,v_3\}$
In other words how can I show that this are ALL the 2 dimensional T invariant subspaces and that there are no other T invariant 2 dimensional subspaces?
Best Answer
Proposition: If $v_1,v_2,..,v_n$ are eigenvectors of $T:V\to V$ corresponding to distinct eigenvalues and $W$ is a T-invariant subspace of V such that $\sum_{i=1}^nv_i\in W$ then $v_i\in W \space \forall i\in\{1,2,..,n\} $.
Proof:Let $\lambda_i$ be the eigenvalue corresponding $v_i$ and I be the identity transformation on $V$.
$$\sum_{i=1}^{m}v_i\in W\Rightarrow (T-\lambda_{m}I)\sum_{i=1}^{m}v_i\in W\text{ (as W is T-invariant)}\Rightarrow \sum_{i=1}^{m-1}(\lambda_{i}-\lambda_{m})v_{i}\in W\\ \Rightarrow (T-\lambda_{m-1}I)\sum_{i=1}^{m-1}(\lambda_{i}-\lambda_{m})v_i\in W\Rightarrow \sum_{i=1}^{m-2}(\lambda_{i}-\lambda_{m})(\lambda_i-\lambda_{m-1})v_{i}\in W\Rightarrow ...\\ \Rightarrow v_1\prod_{i=2}^{m}(\lambda_1-\lambda_i)\in W\Rightarrow v_1\in W\text{ (as $\lambda_i$ are distinct, $\prod_{i=2}^{m}(\lambda_1-\lambda_i)$ is non-zero)}\\ \text{and } \sum_{i=2}^nv_i\in W\;[\;\because\text{ W is a linear subspace}\;]$$ Repeating the same process we see that the proposition is true.
As $v_1,v_2,v_3$ correspond to distinct eigenvalues, they are linearly independent and consequently form a basis for $\mathbb{R}^3$.$v$ is an eigenvector of $T\Rightarrow cv$ is an eigenvector of T corresponding to the same eigenvalue if $c$ is non-zero. So if W is a $2$-dimensional T-invariant subspace and $a,b\neq 0$, $av_i+bv_j\in W\Rightarrow av_i,bv_j\in W\Rightarrow v_i,v_j\in W\Rightarrow W=span\{v_i,v_j\}$.