Question

(c) The method of least-squares states that the vector \(v = (a, b, c)\) that will produce the coefficients for the parabola \(a + bx + cx^2 = y\) that best fits the four data points of part (b) is a solution to the matrix equation \(A^T A v = A^T b\) where \(A\) is the coefficient side of the augmented matrix from part (b) above, and \(b\) is the augmented column of that matrix. (i) (5 points) Find the matrices \(A^T A\) and \(A^T b\) for this matrix equation. (ii) (5 points) Find \((A^T A)^{-1}\), the inverse of \(A^T A\), and use this to solve for \(v\). (iii) (1 point) So what is the parabola \(a + bx + cx^2 = y\) that best fits the four data points of part (b)?

          (c) The method of least-squares states that the vector \(v = (a, b, c)\) that will produce the coefficients for the parabola \(a + bx + cx^2 = y\) that best fits the four data points of part (b) is a solution to the matrix equation
\(A^T A v = A^T b\)
where \(A\) is the coefficient side of the augmented matrix from part (b) above, and \(b\) is the augmented column of that matrix.
(i) (5 points) Find the matrices \(A^T A\) and \(A^T b\) for this matrix equation.
(ii) (5 points) Find \((A^T A)^{-1}\), the inverse of \(A^T A\), and use this to solve for \(v\).
(iii) (1 point) So what is the parabola \(a + bx + cx^2 = y\) that best fits the four data points of part (b)?

(c) The method of least-squares states that the vector v = (a, b, c) that will produce the coefficients for the parabola a + bx + cx^2 = y that best fits the four data points of part (b) is a solution to the matrix equation
A^T A v = A^T b
where A is the coefficient side of the augmented matrix from part (b) above, and b is the augmented column of that matrix.
(i) (5 points) Find the matrices A^T A and A^T b for this matrix equation.
(ii) (5 points) Find (A^T A)^-1, the inverse of A^T A, and use this to solve for v.
(iii) (1 point) So what is the parabola a + bx + cx^2 = y that best fits the four data points of part (b)?

Added by Enrique O.

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