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# Consider the weighted sum least squares objective ||Ax b|| 2 + ||Dx|| 2 , where the n-vector x is the variable, A is an m n matrix, D is the (n 1) n

Consider the weighted sum least squares objective ||Ax − b|| 2 + λ||Dx|| 2 ,

where the n-vector x is the variable, A is an m × n matrix, D is the (n − 1) × n difference matrix, with ith row (ei + 1 − ei) T , and λ > 0. Although it does not matter in this problem, this objective is what we would minimize if we want an x that satisfies Ax ≈ b, and has entries that are smoothly varying. We can express this objective as a standard least squares objective with a stacked matrix of size (m + n − 1) × n. Show that the stacked matrix has linearly independent columns if and only if A1 not equal to 0, i.e., the sum of the columns of A is not zero.