MATH 240 Midterm: MATH 240 NIU Test2Solution
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Be sure to show all necessary work: (25 pts) a = 14 (b) the rank of a is 2; its nullity is 4 2 = 2. (d) basis for the column space of a: columns 1 and 3 of the original matrix. 7 (c) basis for the row space: (1, 1, 0, 3), (0, 0, 1, 2). (e) find a basis for the nullspace of a. The system reduces to this: x1 x2 + 3x4 = 0 and x3 + 2x4 = 0 x1 = x2 3x4 and x3 = 2x4. Letting x2 = 1, x4 = 0 gives. ; letting x2 = 0, x4 = 0 gives. Dependent: the second is a multiple of the rst. Dependent: there are 5 vectors and dim(p3) = 4: (10 pts) find the determinant by reducing to row echelon form. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 5 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12: (10 pts) solve for x: 1 0 0 x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
