Chapter 6.1, Problem 75E

Proof Use the concept of a fixed point of a linear transformation $T:V\to V$. A vector $u$ is a fixed point when $T\left(u\right)=u$.

(a) Prove that $0$ is a fixed point of a liner transformation $T:V\to V$.

(b) Prove that the set of fixed points of a linear transformation $T:V\to V$ is a subspace of $V$.

(c) Determine all fixed points of the linear transformation $T:R^2\to R^2$ represented by $T\left(x,y\right)=\left(x,2y\right)$.

(d) Determine all fixed points of the linear transformation $T:R^2\to R^2$ represented by $T\left(x,y\right)=\left(y,x\right)$.