Question

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If zero is an eigenvalue of a linear transformation T, then T is not invertible.

Answer 1

Answer on Question #44923 – Math - Linear Algebra

If zero is an eigenvalue of a linear transformation $T$, then $T$ is not invertible.

Answer

True. If zero is an eigenvalue of a linear transformation $T$, so there is at least one non-zero vector $\vec{v}$ such that $T\vec{v} = 0$ (0 is an eigenvalue of $T$ with corresponding eigenvector $\vec{v}$). We see that the nullspace of $T$ has dimension $\geq 1$. Since

$$\dim \operatorname{col} T + \dim \operatorname{nul} T = n$$

and

$$\dim \operatorname{col} T = \operatorname{rank} T$$

$\operatorname{rank} T < n$. Then $T$ is not invertible.

www.AssignmentExpert.com