  We present a catalog of 11 multi-planet systems from Campaigns 1 and 2 of the
K2 mission. We report the sizes and orbits of 26 planets split between seven
2-planet systems and four 3-planet systems. These planets stem from a
systematic search of the K2 photometry for all dwarf stars observed by K2 in
these fields. We precisely characterized the host stars with adaptive optics
imaging and analysis of high-resolution optical spectra from Keck/HIRES and
medium-resolution spectra from IRTF/SpeX. We confirm two planet candidates by
mass detection and validate the remaining 24 candidates to $>99\%$ confidence.
Thirteen planets were previously validated or confirmed by other studies and 24
were previously identified as planet candidates. The planets are mostly smaller
than Neptune (21/26 planets) as in the Kepler mission and all have short
periods ($P < 50$ d) due to the duration of the K2 photometry. The host stars
are relatively bright (most have $Kp < 12.5$ mag) and are amenable to follow-up
characterization. For K2-38, we measured precise radial velocities using
Keck/HIRES and provide initial estimates of the planet masses. K2-38b is a
short-period super-Earth with a radius of $1.55 \pm 0.16~R_\oplus$, a mass of
$12.0 \pm 2.9~M_\oplus$, and a high density consistent with an iron-rich
composition. The outer planet K2-38c is a lower density sub-Neptune-size planet
with a radius of $2.42 \pm 0.29~R_\oplus$ and a mass of $9.9 \pm 4.6~M_\oplus$
that likely has a substantial envelope. This new planet sample demonstrates the
capability of K2 to discover numerous planetary systems around bright stars.
