U W u wu U w W R13 have subspace S T dim S 7 dim t 8 a ma

U+ W= {u + w|u U, w W}. R^13 have subspace S, T. dim (S) = 7, dim (t) = 8 (a) max dim (S T) = ? (b) min dim (S T) = ? (c) max dim (S + T) = ? (d) min dim (S + T) = ? (e) dim (S T) + dim (S + T) = ? 38. Consider the bases S= {u_1,u_2,u_3 } and T= {v_1,v_2,v_3}, with u_1 = [-3 0 -3], u_2 = [-3 2 -1], u_3 = [1 6 -1], v_1 = [-6 -6 0], v_2 = [-2 -6 4], v_3 = [-2 -3 7]. (a) Find the transition matrix from S to T. (b) Using the result in (a), compute the coordinate vector [w]_T where w = [-5 8 -5].

Solution

a) max of 7 and 8 = 8

b) min of 7 and 8 = 7

c) max S+T= 7+8 =15

d) min S+T = max of 7 or 8 = 8

e) dim ST + dim(S+T) = 13