We measure the filling factor, correlation function, and power spectrum of transmitted flux in a large sample of Lyα forest spectra, comprised of 30 Keck HIRES spectra and 23 Keck LRIS spectra. We infer the linear matter power spectrum P(k) from the flux power spectrum PF(k), using an improved version of the method of Croft et al. that accounts for the influence of redshift-space distortions, nonlinearity, and thermal broadening on the shape of PF(k). The evolution of the shape and amplitude of P(k) over the redshift range of the sample (z 2-4) is consistent with the predictions of gravitational instability, implying that nongravitational fluctuations do not make a large contribution to structure in the Lyα forest. Our fiducial measurement of P(k) comes from a subset of the data with 2.3 < z < 2.9, mean absorption redshift z = 2.72, and total path length Δz 25. It has a dimensionless amplitude Δ2(kp) = 0.74 at wavenumber kp = 0.03 (km s-1)-1 and is well described by a power law of index ν = -2.43 ± 0.06 or by a CDM-like power spectrum with shape parameter Γ' = 1.3 × 10-3 (km s-1)-1 at z = 2.72 (all error bars 1 σ). The correspondence to present-day P(k) parameters depends on the adopted cosmology. For Ωm = 0.4, ΩΛ = 0.6, the best-fit shape parameter is Γ = 0.16 h Mpc-1, in good agreement with measurements from the 2dF Galaxy Redshift Survey, and the best-fit normalization is σ8 = 0.82(Γ/0.15)-0.44. Matching the observed cluster mass function and our measured Δ2(kp) in spatially flat cosmological models requires Ωm = 0.38 + 2.2(Γ - 0.15). Matching Δ2(kp) in COBE-normalized, flat CDM models with no tensor fluctuations requires Ωm = (0.29 ± 0.04)n-2.89 h, and models that satisfy this constraint are also consistent with our measured logarithmic slope. The Lyα forest complements other observational probes of the linear matter power spectrum by constraining a regime of redshift and length scale not accessible by other means, and the consistency of these inferred parameters with independent estimates provides further support for a cosmological model based on inflation, cold dark matter, and vacuum energy.