  We present spectroscopic observations of the host galaxies of 82 low-redshift
type Ia supernovae (SNe Ia) discovered by the Palomar Transient Factory (PTF).
We determine star-formation rates, gas-phase/stellar metallicities, and stellar
masses and ages of these objects. As expected, strong correlations between the
SN Ia light-curve width (stretch) and the host age/mass/metallicity are found:
fainter, faster-declining events tend to be hosted by older/massive/metal-rich
galaxies. There is some evidence that redder SNe Ia explode in higher
metallicity galaxies, but we found no relation between the SN colour and host
galaxy extinction based on the Balmer decrement, suggesting that the colour
variation of these SNe does not primarily arise from this source. SNe Ia in
higher-mass/metallicity galaxies also appear brighter after stretch/colour
corrections than their counterparts in lower mass hosts, and the stronger
correlation is with gas-phase metallicity suggesting this may be the more
important variable. We also compared the host stellar mass distribution to that
in galaxy targeted SN surveys and the high-redshift untargeted Supernova Legacy
Survey (SNLS). SNLS has many more low mass galaxies, while the targeted
searches have fewer. This can be explained by an evolution in the galaxy
stellar mass function, coupled with a SN delay-time distribution proportional
to $t^{-1}$. Finally, we found no significant difference in the
mass--metallicity relation of our SN Ia hosts compared to field galaxies,
suggesting any metallicity effect on the SN Ia rate is small.
