2. Note that the following four subquestions are independent of each other. (a) [10 points] Consider the following three

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2 Note That The Following Four Subquestions Are Independent Of Each Other A 10 Points Consider The Following Three 1 (78.87 KiB) Viewed 27 times

2. Note that the following four subquestions are independent of each other. (a) [10 points] Consider the following three vectors in R3 : 2 2 -4 ai = a2 = a3 = 9 -8 12 4 2 8 6 0 Is a Espan{a2, a3}? Is 0 E span{az, a3}, where 0 = 0 ? Justify your answers. 0 (b) [8 points) Suppose {bı, b2} is a linearly independent set in R7. Show that {bı 7b2, 7b2 + b } is also a linearly indepedent set. Is bị a linear combination of bi - 7b2 and 7b2 + b1 ? Justify your answers. (c) [5 points] Consider the real vector space R4. Let ci = (2,-1,-1,-1), C2 = (1, 2, -1, -3), C3 = (-1,3,0, -2) Determine whether C1, C2, and c3 are linearly dependent. Find the dimension and a basis for the subspace span {C1, C2, C3}. (d) [7 points) Suppose that W = {41, 42, ..., Yn} is a basis for R". Show that if M is an n xn invertible matrix, then R= {My1, My2, ..., Myn} is linearly independent.