Games have shown to be great tools for learning a variety of topics, but few games have been developed mathematics topics at the undergraduate level. One possible avenue is to pursue digital game-based learning. The game "Vector Unknown" was designed from a task from the Inquiry-Oriented Linear Algebra (IOLA) curriculum to connect student's algebraic and geometric understandings of vector equations in linear algebra. In the game, users guide an avatar to a goal location by selecting vectors and scaling them. This research utilizes the game as realistic basis for supporting student understanding of 2D space structured by linear combinations, which may provide crucial intuitions between concepts such as span, linear independence, basis, and abstract vector spaces. Six participants were chosen from a large Southern university with a variety of gaming experience and no linear algebra experiences. The participants were paired into three groups based upon game experience (a gamer/gamer group, a gamer/non-gamer group, and a non-gamer/non-gamer group). Participants played and analyzed the game "Vector Unknown" to understand its mathematical design over the course of two approximately 60-minute interviews. During a third 60-minute interview, participants were then asked to design their own version of 3D version of the game to see how their reasoning about linear combinations in 2D space extends to higher dimensions. I conducted an analysis of the student's activity and found that students had four main themes for structuring space including structuring by design, structuring by generating, structuring across dimensions, and structuring all possible goals. I also found that viewpoint played an important role in student's connecting 2D and 3D space and include a viewpoint analysis linking differing viewpoints in GeoGebra to student activity. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]