How to keep value of checkbox in dataGridView after re-bind

I have a dataGridView that is populated by a query binding to DataSource. The DGV has a first column as a checkboxcolumn so the user can choose which products he wants to order.

I have also created a "search" functionality (a simple textbox and a button). When the user clicks the button, in the click_event, I redefine the query (something like "...where Name like '%searchword%'"). The search works fine and it filters correctly. The problem is that the checkbox selection loses its value when I re-bind the DGV.

I want to know how I can keep track of the checkbox values.

I tried to put the id's on a list, but I don't know how to handles this if, for example, the user unselects the box (how would I delete it from the list).

Here's the code:

    public Main()
    {
        InitializeComponent();
        SendMessage(textBoxSearch.Handle, 0x1501, 1, "Search Product by Name...");

        try
        {
            SQLiteConnection con = new SQLiteConnection(conString);

            string prod = "select id,Name,Bulk_Density,Max_Size,Min_Size,WP,GS from Product order by Name";
            SQLiteDataAdapter da = new SQLiteDataAdapter(prod, con);
            con.Open();
            DataSet ds = new DataSet();
            da.Fill(ds, "PROD");
            dataGridViewProduct.DataSource = ds.Tables["PROD"];



            dataGridViewProduct.Columns[1].Visible= false;
            dataGridViewProduct.Columns[2].HeaderText = "Name";
            dataGridViewProduct.Columns[3].HeaderText = "Bulk Density [g/l]";
            dataGridViewProduct.Columns[4].HeaderText = "Max Size [mm]";
            dataGridViewProduct.Columns[5].HeaderText = "Min Size [mm]";
            dataGridViewProduct.Columns[2].ReadOnly = true;
            dataGridViewProduct.Columns[3].ReadOnly = true;
            dataGridViewProduct.Columns[4].ReadOnly = true;
            dataGridViewProduct.Columns[5].ReadOnly = true;
            dataGridViewProduct.Columns[6].ReadOnly = true;
            dataGridViewProduct.Columns[7].ReadOnly = true;
            dataGridViewProduct.Columns[2].SortMode = DataGridViewColumnSortMode.NotSortable;
            dataGridViewProduct.Columns[3].SortMode = DataGridViewColumnSortMode.NotSortable;
            dataGridViewProduct.Columns[4].SortMode = DataGridViewColumnSortMode.NotSortable;
            dataGridViewProduct.Columns[5].SortMode = DataGridViewColumnSortMode.NotSortable;
            dataGridViewProduct.Columns[6].SortMode = DataGridViewColumnSortMode.NotSortable;
            dataGridViewProduct.Columns[7].SortMode = DataGridViewColumnSortMode.NotSortable;


            dataGridViewProduct.ColumnHeadersDefaultCellStyle.Alignment = DataGridViewContentAlignment.MiddleCenter;
            con.Close();
        }
        catch (SQLiteException sqle)
        {
            MessageBox.Show(sqle.Message);
        }
    }

    private void buttonSearch_Click(object sender, EventArgs e)
    {
        SQLiteConnection con = new SQLiteConnection(conString);

        string search = textBoxSearch.Text;
        string prod = "select id,Name,Bulk_Density,Max_Size,Min_Size,WP,GS from Product where Name like '%"+search+"%' order by Name";
        SQLiteDataAdapter da = new SQLiteDataAdapter(prod, con);
        con.Open();
        DataSet ds = new DataSet();
        da.Fill(ds, "PROD");
        dataGridViewProduct.DataSource = ds.Tables["PROD"];
    }