for test, create some image files

dd if=/dev/zero of=file1.img bs=1M count=0 seek=5120
dd if=/dev/zero of=file2.img bs=1M count=0 seek=5120
losetup /dev/loop1 file1.img
losetup /dev/loop2 file2.img

create and remove pv

pvcreate /dev/loop1
pvs
pvdisplay
pvdisplay -m
pvremove /dev/loop1

pvmove can move extents from one pv to another

pvmove /dev/loop1

create and vemove vg

vgcreate vgname /dev/loop1
vgs
vgdisplay
vgchange -an vgname
(if you remove the disk and insert it to another system,
you can find the volume group unless you remove the volume group)
vgremove vgname

you can add/remove pv to vg

vgextend vgname /dev/loop2
vgreduce vgname /dev/loop1

create logical volume

lvcreate -L 1G -n lvname vgname
lvs
lvdisplay
mkfs -t ext4 /dev/vgname/lvname
mount /dev/vgname/lvname /mountpoint

mirror or stripe

lvcreate -L 128M -m 1 -n lvname vgname
lvcreate -L 128M -i 2 -n lvname vgname

snapshot and merge. if you don't want to merge, you can simply remove the snapshot lv

lvcreate -L 64M --snapshot /dev/vgname/lvname -n lvsnap
lvconvert --merge vgname/lvsnap

extend logical volume and filesystem

lvextend -L +512MB /dev/vgname/lvname
e2fsck -f /dev/vgname/lvname
resize2fs /dev/vgname/lvname

shrink filesystem and logical volume

e2fsck -f /dev/vgname/lvname
resize2fs /dev/vgname/lvname 500M
lvchange -an /dev/vgname/lvname
lvreduce -L -512MB /dev/vgname/lvname
lvchange -ay /dev/vgname/lvname
resize2fs /dev/vgname/lvname

remove logical volume

lvchange -an /dev/vgname/lvname
lvremove /dev/vgname/lvname

umount

fuser -vm /mountpoint
umount /mountpoint

extend pv

parted /dev/sda
parted /dev/sda print
parted /dev/sda rm 2
parted /dev/sda resizepart 1 100%

pvdisplay /dev/sda1
vgdisplay vgname
pvresize /dev/sda1

extend partition

parted /dev/sdx print
parted /dev/sdx resizepart [num] 100%
e2fsck -f /dev/sdx[num]
resize2fs /dev/sdx[num]