This paper proposes two local exponential observers for left-invariant systems on linear Lie groups, where the full state of the system is available for measurement. We show that, depending on the observer chosen, local exponential stability of one of the two estimation error dynamics, left or right invariant error dynamics, is obtained. Our proposed observers are noteworthy because their estimation error dynamics are differentially equivalent to a linear and stable differential equation on the Lie algebra. We illustrate our observer designs for an attitude estimation problem on the special orthogonal group SO(3).