DataGridView bound to DataTable (display & update problem with checkbox column type)

[exterminator]

( Visual C# 2008, .Net framework 3.5 sp1)

The title line should be explanatory but still the details:

1. I have a DataGridView control on a Window Form.
2. AutoGenerateColumns property set to False.
3. Columns added in design time and each columns' DataPropertyName set to specific column name in DataTable.
4. DataTable populated as:

   DataTable dataTable = new DataTable();
   OleDbConnection conn = DBConnection.getConnection();
   string ct = @"SELECT column1, column2 FROM table";
   OleDbDataAdapter da = new OleDbDataAdapter(ct, conn);
   OleDbCommandBuilder cb = new OleDbCommandBuilder(da);
   da.Fill(dataTable);

(Feel free to suggest any corrections in code above, if something looks dodgy)
5. column 2 is supposed to be a boolean/a flag. So, I choose 2nd column in datagridview to be of check-box type type (at design time).

There are 2 problems:
a) What do I need to do so that a value true in DB for column 2 shows check-box as ticked (it isn't as of now while displaying data)?
b) What do I need to do to have the data updated into DB?

a) might help me find answer to b) but currently, I am doing the following for b) which doesn't seem to be working. I came across variety of expcetions that I searched for and one thing led to another but still I seem far from the solution. I will just throw it infront of you to help me how to achieve what I want to. This is just displaying a two column table on the Form in DataGridView and saving any updates on 2nd column value into the DB.

  string ct = @"SELECT column1, column2 FROM table";
  OleDbConnection conn = DBConnection.getConnection();
  OleDbDataAdapter da = new OleDbDataAdapter(ct, conn);
  OleDbCommandBuilder cb = new OleDbCommandBuilder(da);
  da.MissingSchemaAction = MissingSchemaAction.AddWithKey;
  /*
    da.UpdateCommand = new OleDbCommand();
    da.UpdateCommand.CommandText =@"update table set column2 = ? where column1 = ?";
    da.UpdateCommand.Connection = conn;
    OleDbParameter p1 = da.UpdateCommand.Parameters.Add("@p1", OleDbType.Integer);
    p1.SourceColumn = "column2";
    p1.SourceVersion = DataRowVersion.Original;
    OleDbParameter p2 = da.UpdateCommand.Parameters.Add("@p2", OleDbType.VarChar);
    p2.SourceColumn = "column1";
    p2.SourceVersion = DataRowVersion.Original;
  */
  da.Update(dataTable);

I commented out some of the code above since it didn't seem to work (propagate the updates to the db). I had added it since otherwise I was getting the exception: "Dynamic SQL generation for the UpdateCommand is not supported against a SelectCommand that does not return any key column information." (There are no keys in table at the moment, and I don't need one too).

Any suggestions how to achieve the above?