Let X be the design matrix used to find the least-squares line to fit data (x₁y₁),.....(x_n,y_n). Use a theorem in Section 6.5 to show that the normal equations have a unique solution if and only if the data include at least two data points with different x-coordinates.

Data From Section 6.5

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Let A be an m x n matrix. The following statements are logically equivalent: a. The equation Ax = b has a unique least-squares solution for each b in R™. b. The columns of A are linearly independent. c. The matrix ATA is invertible. When these statements are true, the least-squares solution x is given by x = (A¹A)-¹A¹b