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:

  To be linearly independent means that the only solution to the system equaling the zero matrix is the trivial solution (meaning all 0's) so for a) you need to show that for a1*(1,2,2,1) + a2*(0,2,0,1) + a3(-2,0,-4,3) = (0,0,0,0), there is NO other solution other than a1,a2,a3 = 0. for b) You can tell that v is a linear combination if there is a solution to the vector equation a1(v_1) + a2(v_2) + a3(v_3) = v if there is no solution, then it is not a linear combination, and they do not span. Hope this helps!