Some student-created multiple choice questions, with solutions and explanations provided.
- Let $(v_1, v_2, v_3)$ be a basis for $\mathbb R^3$. If $T \in \mathcal L(\mathbb R^3)$ is such that $Tv_i = 0$ for $i=1,2,3$, how many linearly independent eigenvectors does $T$ have?
- (A) 0
- (B) 1
- (C) 2
- (D) 3
- (E) Impossible to determine without further information
- ANSWER: (D). $T$ is clearly the zero operator, for which every vector in $\mathbb R^3$ is an eigenvector. Since the dimension is 3, there are 3 linearly independent eigenvectors.