Thermal expansion is basically the spreading out of substances when they get heated. The atoms get "excited" as they gain heat energy, and therefore move away from each other, expanding.
Linear thermal expansion is the same concept, but the expansion makes the item longer- if we want to measure linear expansion, we measure the length by which something has expanded (or retracted) due to gaining (or losing) heat.
For example, lets think about a thin metal rod that has gained a certain amount of energy. It will expand linearly, meaning that the rod will become slightly longer. To calculate the change in the length, we use this equation:
# Delta L= Lalpha Delta T #
With #DeltaL# = Change in length.
#L=# initial length.
# alpha#=the coefficient of linear expansion (this is normally given, though if given the value of the other variables, can be found!)
#Delta T#= Change in temperature
Note that this equation gives the change in linear expansion and not the new length. If the object was expanding, then we add the #Delta L# to the initial length #L#