Consider the set of vectors in R 5 ; B1 = {w1, w2, w3, w4, w5, w6} where w1 = (1 1 2 4 1 ) w2 = ( 1 1 2 1 1 ) w3 = ( 3 1 1 2 0 ) w4 = ( 2 1 0 3 1 ) w5 = ( 5 4 1 11 3 ) , w6 = (1 0 1 2 1 )

Show B1 is a linearly dependent set. Then, demonstrate the conclusion of Theorem 1.2.2: Find a maximal linearly independent set B 1 of vectors from B1, and show that the vectors from B1 that are NOT in B 1 set are contained in the span of B 1 (and hence, that span B1 = span B 1 ).

What is the dimension of span B1?

Consider the set B2 = {z1, z2, z3, z4, z5} where z1 = (5 2 1 7 1 ) , z2 = ( 2 1 0 0 1 ) , z3 = (1 2 1 1 0 ) , z4 = ( 2 4 2 4 1 ) , z5 = (0 1 2 3 1)

Find all the vectors in the intersection span B1 span B2. Show that this is a subspace.