Given x1 = (1,1,1)T and x2 = (3,−1,4)T: (a) Do x1 and x2 span R3 ? Explain. (b) Let x3 be a third vector in R3and set X = (x1 x2 x3). What condition(s) would X have to satisfy in order for x1, x2, and x3 to form a basis for R3?

2. Consider the following sets of vectors. Show that each set (i) contains the zero vector, (ii) is closed under addition and scalar multiplication. Then find a basis for each set and give the dimension. (a) W is the set of vectors of the form (3s, -2s, s). (b) W is the set of vectors of the form (t – 2, 0,6 – 3t). (c) H is the set of vectors of the form (2b, a, 3b) (d) H is the set of vectors of the form (-b+ 2c, b, 4c),

4. Let S = {v,,v2, V3, V4, V5} where, 3 1 2 3 3 ,V2 9. ,V3 2 la -1 4 Find a basis for the space span(S) which is a subset of S. Then express each vector that is not in the basis as a linear combination of the basis vectors.

1. Determine whether the vectors (1,-1) and (1, 1)T are linearly inde- pendent or not. 2. Show that the vectors (1,0,0), (1, 1, 0) and (1,1,1) form a basis of R³. 3. Suppose that the vectors (1, 2, -1,0) and (1,3,2,0) span the sub- space U of R4. Determine whether the vector (1, 1, 1, 0) belongs to U or not.