The cognitive structure of an algebra test was defined and validated using the linear test model (LLTM) and quadratic assignment (QA) techniques. The LLTM is an extension of the Rasch model with a linear constraint that describes item difficulty in terms of the cognitive operations required to solve the item. The model permits the specification of the cognitive structure of a test via the weight matrix. In this study, the structure was based on eight production rules that represent the mathematical procedures for solving linear equations with one variable. A 29-item test was constructed and administered to 235 ninth graders in April of 1991. Subjects included 94 low scorers and 141 high scorers. Item response data were analyzed using the LLTM computer program. Estimates of the basic parameters corresponding to each production rule were obtained. A QA confirmatory approach validated the cognitive structure of the test. The proximity matrix represented the examinees' actual use of production rules in solving every pair of items. Cell entries of the structure matrix reflected the common number of production rules required to solve every item pair. The cognitive structure of the test was validated (i.e., examinees solved items using the rules specified in the weight matrix). This supports the basic parameters estimated via the LLTM as quantitative indexes of the difficulty of the cognitive rules in the weight matrix. One figure and seven tables provide study data. There is a 35-item list of references. (Author/SLD)