How many Hecke operators span the Hecke algebra?

Linear Algebra Question (Linear independence, subspace vectors span, linear combination)?

• Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination of v_1, v_2, and v_3? Please explain all steps. Thank you!

• Answer:

  a) As given vectors will not form square matrix ,so maximum rank will be 3 if rank of matrix =3( max. possible rank) then vectors will be L.I if rank of matrix <3 then vectors will be L.D matrix form will be 1 2 2 1 0 2 0 1 -2 0 -4 3 we will try to reduce in normal form 1 2 2 1 0 2 0 1 0 4 0 5 1 0 0 0 0 2 0 1 0 4 0 5 1 0 0 0 0 1 0 1 C2-->(C2)/2 0 2 0 5 1 0 0 0 0 1 0 1 0 0 0 3 1 0 0 0 0 1 0 0 0 0 0 3 1 0 0 0 0 1 0 0 0 0 1 0 which [I3,0] form rank =3 hence L.I