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There is little doubt that linear algebra is a fundamentally important course in undergraduate studies. It is required for students majoring not only in mathematics, but also in engineering, physics, and economics, to name just a few. However, to date, research on students' understanding of linear algebra is rather slim. This study is a contribution to the ongoing research in undergraduate mathematics education, focusing on linear algebra. It is guided by the belief that better understanding of students' difficulties leads to improved instructional methods. The questions posed in this study are: What is students' understanding of the key concepts of linear algebra? What difficulties do students experience when engaged in these tasks? What can example-generation tasks reveal about students' understanding of linear algebra? This study identifies some of the difficulties experienced by students with learning several key concepts of linear algebra, and also isolates some possible obstacles to such learning. In addition, this study introduces learner-generated examples as a pedagogical tool that helps learners partly overcome these obstacles.