- Note That The Following Four Subquestions Are Independent Of Each Other A 10 Points Consider The Following Three Ve 1 (29.81 KiB) Viewed 18 times
Note that the following four subquestions are independent of each other. (a) [10 points] Consider the following three vectors in R: ai - (3) a2 = 4 a3 = 2 8 12 4 6 0 Is ai Espan{az, az}? Is 0 Espan{a2, a3}, where 0 = (8) 0 ? Justify your answers. 0 a (b) 18 points) Suppose {bu, b2} is a linearly independent set in RT. Show that {bı - 7b2,752 + b } is also a linearly indepedent set. Is by a linear combination of b1 - 7b2 and 7b2 + bu ? Justify your answers. (c) (5 points] Consider the real vector space 4. Let C1 = (2,-1,-1,-1), c2 = (1,2,-1,-3). c3 =(-1,3,0,-2) Determine whether C1.C2, and ca are linearly dependent. Find the dimension and a basis for the Subspace span {c1, c2,c3}. (d) 17 points] Suppose that W = {91, V2, ..., yn} is a basis for R". Show that if M is an n xn invertible matrix, then R = {Myı, My2, ...,Myn} is linearly independent. a