I was trying to solve the following problem from a competitive exam paper.
Let $A$ be a nonzero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be the image of V under A. Let $k=dimV_0 < n$ and suppose for some $\lambda \in \mathbb R$ , $A^2=\lambda A$. Then
- $\lambda = 1$
- $detA=|\lambda|^n$
- $\lambda$ is the only eigen value of $A$
- There is a non trivial subspace $V_1 \subset V$ such that $Ax=0\forall x \in V_1$
One or more options may be correct.
What I think is that option $4$ is correct as Rank($A$)=$dimV_0=k$ So by Rank-Nullity Theorem $nullity(A)=n-k>0$ Thus nullspace of $A$ is nontrivial.
Am I right? Also please help me about the other options.
Thnx in advance
Best Answer
For $V = \mathbb{R}^3$, with the canonical basis $(1,0,0),(0,1,0),(0,0,1)$ let $A$ be the map corresponding to the matrix $$ M_A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} \text{.} $$
Then $V_0 = \textrm{span }\{(0,1,0),(0,0,1)\}$, and $A^2 = 2A$.
This counter-example disproves 1., 2. and 3.
And 4. is indeed true, for the reason you stated.