Question

Let U be the set of all 2x2 upper triangular matrices with real entries show that B-{[6] [8]} is a linearly indepandewe set mo Explain why B is not a basis for U Include one more matry in B so that this becames a basis for U

Answer 1

Given set B= {lo :] [: 8}} is subset of all des upper 2 trangular matrices ci) To show that, B is L.I there exist scolar a b such that 9 [:]++(21-69 lo J+6 23.600) atb 9. ů en] 9Ь T atb=0, a=, 2b = 0 a=o b=0 = Henee, set cii) B is lin early independent. since, demension of all exa upper triangular {[53,70 !)-C01]} B contains anly of all exe upper triangular matrices is 3 generated by sel l.e. hore ser Henee B is anley e elements basis of u not ci) Now odd then B = {[J[:] [:]} is L.I. and [oo in B generate U. B is Hlence Basis of U.