Applications: Least Squares and Projections
When you encounter a system of linear equations with more equations than unknowns, you face an overdetermined system. Such systems often have no exact solution, so instead you seek the "best" approximate solution. The least squares approach is a standard method for this, minimizing the sum of the squares of the residuals (the differences between the observed and predicted values). Mathematically, for a given matrix A (of size m by n, with m>n) and a vector b in Rm, you want to find the vector x in Rn that minimizes the norm ∣∣Ax−b∣∣2. This is equivalent to projecting b onto the column space of A. Geometrically, the least squares solution is the point in the column space of A closest to b, making the error vector b−Ax orthogonal to every column of A.
123456789101112131415# Define an overdetermined system: 3 equations, 2 unknowns A <- matrix(c(1, 1, 1, 2, 1, 3), nrow = 3, byrow = TRUE) b <- c(1, 2, 2) # Solve the least squares problem using QR decomposition x_ls <- qr.solve(A, b) # Print the least squares solution print(x_ls) # Compute the projection of b onto the column space of A b_proj <- A %*% x_ls print(b_proj)
In the code above, you use the qr.solve() function to compute the least squares solution to an overdetermined system. The solution vector x_ls minimizes the squared distance between Ax and b. The product A %*% x_ls gives the projection of b onto the column space of A, which represents the closest point in that subspace to b. This projection is fundamental in linear algebra and optimization, as it provides the best approximation to b using a linear combination of the columns of A. The error vector b - A %*% x_ls is orthogonal to the column space of A, confirming the geometric interpretation of least squares as orthogonal projection.
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Can you explain how the QR decomposition helps solve the least squares problem?
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When you encounter a system of linear equations with more equations than unknowns, you face an overdetermined system. Such systems often have no exact solution, so instead you seek the "best" approximate solution. The least squares approach is a standard method for this, minimizing the sum of the squares of the residuals (the differences between the observed and predicted values). Mathematically, for a given matrix A (of size m by n, with m>n) and a vector b in Rm, you want to find the vector x in Rn that minimizes the norm ∣∣Ax−b∣∣2. This is equivalent to projecting b onto the column space of A. Geometrically, the least squares solution is the point in the column space of A closest to b, making the error vector b−Ax orthogonal to every column of A.
123456789101112131415# Define an overdetermined system: 3 equations, 2 unknowns A <- matrix(c(1, 1, 1, 2, 1, 3), nrow = 3, byrow = TRUE) b <- c(1, 2, 2) # Solve the least squares problem using QR decomposition x_ls <- qr.solve(A, b) # Print the least squares solution print(x_ls) # Compute the projection of b onto the column space of A b_proj <- A %*% x_ls print(b_proj)
In the code above, you use the qr.solve() function to compute the least squares solution to an overdetermined system. The solution vector x_ls minimizes the squared distance between Ax and b. The product A %*% x_ls gives the projection of b onto the column space of A, which represents the closest point in that subspace to b. This projection is fundamental in linear algebra and optimization, as it provides the best approximation to b using a linear combination of the columns of A. The error vector b - A %*% x_ls is orthogonal to the column space of A, confirming the geometric interpretation of least squares as orthogonal projection.
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