A Note on Background Independence in Noncommutative Gauge Theories, Matrix Model and Tachyon Condensation

Using the construction of D-branes with nonzero $B$ field in the matrix model we give a physical interpretation of the known background independence in gauge theories on a noncommutative space. The background independent variables are identified as the degrees of freedom of the underlying matrix model. This clarifies and extends some recent results about the end point of tachyon condensation in D-branes with a $B$ field. We also explain the freedom in the description which is parametrized by a two form $\mathrm{\Phi }$ from the points of view of the noncommutative geometry on the worldvolume of the branes, and of the first quantized string theory.