Let $T : \mathbb{R}^n → \mathbb{R}^n$ be a linear transformation of $\mathbb{R}^n$
, where $n ≥ 3$, and let $λ_1, . . . , λ_n ∈ \mathbb{C}$ be the eigenvalues of T. Which of the following statements are true?
(a) If $λ_i = 0$, for some $i = 1, . . . , n$, then $T$ is not surjective.
(b) If $T$ is injective, then $λ_i = 1$ for some $i, 1 ≤ i ≤ n$.
(c) If there is a $3$-dimensional subspace $U$ of $V$ such that $T(U) = U$, then $λ_i ∈ \mathbb{R}$ for some $i$, $1 ≤ i ≤ n$.
(a) the product of the eigen values= the determinant of a matrix.
If one eigen value is zero then determinant will be zero and hence the rank of matrix will be the less then the dimension of the space and hence the map is not surjective.
so it is true.
(b)it is false obviously.
(c) But I am not sure here.
how can I able to tackle it?
Best Answer
The restriction of $T$ on $U$ is an endomorphism since $U$ is invariant by $T$ and its characteristic polynomial with degree $3$ has a real root (eigenvalue) by the intermediate value theorem hence the answer is YES.