UIN: KEY Problem 4. (25 points) Suppose that a1, a2, a3, d4, a5, 06, a7 are vectors in R* such that the 4 x 7 matrix A = [al ... a7] that has these vectors as its columns has the following reduced echelon form: -3 0 R = RREF(A) = 0 6 (a) (8 points) Write a5 as a linear combination of a1, a2, a3, and a4. (Since you do not actually know the entries of these vectors, your answer should be in terms of the as with coefficients in front.) = 20 + 3 0 - 4a ( use the coefficients 70 5 from the sth column of the RREF Za + 0a + 3a - 4a (b) (3 points) Which of the vectors a1, . .., a7 cannot be written as linear combinations of a1, d2, d3, and a4? (No justification necessary.) a and a 7 (c) (4 points) What are dim col(A) and dim null(A)? (No justification necessary.) 17 4 3 ( 3 non - pivot columns ) ( 4 pivot columns ) (d) (10 points) Consider the two sets of vectors, {a2, a3, d4, as} and {al, d4, 96, a7}. Exactly one of these two sets is linearly dependent. Identify which one it is, and write down a nontrivial dependence relation among the vectors in it. L. I : none can be made L. D .: a can be using the others made using only a a and a 3 a a a is dependent. Since a = -a+ sa + ba 6 / a nontrivial dependence relation is a - 5 a - ba + a_ = 0 7