change column value with arthimatic sequences using df.loc in pandas

suppose I have following dataframe :

data = {"age":[2,3,2,5,9,12,20,43,55,60],'alpha' : [0,0,0,0,0,0,0,0,0,0]}
df = pd.DataFrame(data)

I want to change value of column alpha based on column age using df.loc and an arithmetic sequences but I got syntax error:

df.loc[((df.age <=4)) , "alpha"] = ".4"          
df.loc[((df.age >= 5)) & ((df.age <= 20)), "alpha"] = 0.4 + (1 - 0.4)*((df$age - 4)/(20 - 4))
df.loc[((df.age > 20)) , "alpha"] = "1"

thank you in davance.

>Solution :

Reference the age column using a . not a $

df.loc[((df.age >= 5)) & ((df.age <= 20)), "alpha"] = 0.4 + (1 - 0.4)*((df.age - 4)/(20 - 4))