A new approach to the classification of QSO rest-frame optical and ultraviolet spectra is presented. The statistical technique of principal component analysis (PCA) is applied directly to a sample of QSO ultraviolet spectra, rest-frame wavelengths lambdalambda1150-2000, taken from the Large, Bright QSO Survey. A detailed discussion of the application of the PCA technique to QSO spectra is given. The PCA approach provides new insights into the relation between the continuum, emission-line, and broad-absorption-line properties of QSOs. The QSO spectra can be described using a small number of principal components, and a quantitative classification scheme for the rest-frame ultraviolet region has been devised. The major results of the analysis can be summarized as follows: 1. The first three principal components obtained from the PCA account for approximately 75% of the QSO sample intrinsic variance, and a quantitative classification scheme based on the contribution of the components to each QSO spectrum can be developed. 2. One of the components describes the emission-line core strength, the second the continuum slope, and the third the broad absorption features present in a subset of the spectra. 3. Broad-absorption-line QSOs are naturally identified as a subclass of the QSO population. They show excess N V and Al III + C III] emission relative to the QSO population as a whole. 4. The emission lines of the QSO sample show a spread of properties which can be arranged in a sequence ranging from broad, low-equivalent-width lines to narrow, high-equivalent-width lines. Other properties of the QSO spectra also vary along this sequence. For example, QSOs in the former category have higher N V/Lyman alpha, N V/C IV, and C IV/Lyman-alpha line ratios than objects with lines characteristic of the latter category, and the wavelength separation between the Lyman-alpha and C IV emission lines is larger in spectra in the former category than in the latter. 5. The continuum hardness correlates with the equivalent widths of C IV and Lyman-alpha emission lines but with a large scatter, and the equivalent width of the Al III/C III] blend correlate with a smaller scatter. 6. A Baldwin Effect consistent with the results of other studies is present in the LBQS sample.