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diff --git a/examples/main.py b/examples/main.py
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+import sympy
+import polymatrix
+
+state = polymatrix.init_expression_state()
+
+x1, x2 = sympy.symbols('x1, x2')
+x = polymatrix.from_((x1, x2))
+
+f1 = polymatrix.from_(x1 + x2)
+f2 = polymatrix.from_(x1 + x1*x2)
+
+f3 = f1 + f2          
+
+# prints the data structure of the expression
+# AdditionExprImpl(left=FromSympyExprImpl(data=((x1 + x2,),)), right=FromSympyExprImpl(data=((x1*x2 + x1,),)))
+print(f3)
+
+state, poly_matrix = f3.apply(state)
+
+# prints the data structure of the polynomial matrix
+# PolyMatrixImpl(terms={(0, 0): {((0, 1), (1, 1)): 1, ((0, 1),): 2, ((1, 1),): 1}}, shape=(1, 1))
+print(poly_matrix)
+
+state, sympy_repr = polymatrix.to_sympy(f3,).apply(state)
+
+# prints the sympy representation of the polynomial matrix
+# [[x1*x2 + 2*x1 + x2]]
+print(sympy_repr)
+
+state, dense_repr = polymatrix.to_dense((f3,), x).apply(state)
+
+# prints the numpy matrix representations of the polynomial matrix
+# array([[2., 1.]])
+# array([[0. , 0.5, 0.5, 0. ]])
+print(dense_repr.data[0][1])               # numpy array
+print(dense_repr.data[0][2].toarray())     # sparse scipy array
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