Let A: = 1 1 2 011 0 0 | (a) Find a basis for ran(A). What is dim(ran(A))? (b) Use the Rank-Nullity theorem to find dim (ker (A)). (c) Find a basis for ker(A). (d) Use the basis from (c) to write the solution sets to each of the following: Ax = 0 Ay= |1 Az= Solution. Let ₁ be the first column of A, 2 the second, and a3 the third column of A. (a) Since σ ₁ and 2 are not multiples of one another, they are linearly independent, but ã₁ + a2 = ã3 and so a basis for ran(A) is {ā₁, ā2} and so dim(ran(A)) = 2. (b) The rank-nullity theorem gives 3 = dim(ran(A)) + dim(ker(A)) and since dim(ran(A)) = 2, dim(ker(A)) = 1. (c) Since dim(ker(A)) = 1, we just need one nonzero element of the kernel. Since a1a2a3 = 0, so a basis for ker(A) is (1, 1, −1). = = α1 + α2 − α3 = 0