Find the least squares solutions to the following linear systems: (a) x+2 y=1,3 x-y=0,-x+2 y=3,

step 1 Here we have a system of three equations with four variables. Because there are four variables a

Find the least squares solutions to the following linear...

Key Concepts

Least Squares Approximation

The least squares method is a strategy to find an approximate solution to an overdetermined system (a system where there are more equations than unknowns) by minimizing the sum of the squares of the residuals—the differences between the observed values and those predicted by the model. This approach provides the best possible solution in terms of minimizing the overall error when an exact solution does not exist.

Overdetermined Systems

Overdetermined systems are systems of linear equations that have more equations than unknowns, which usually leads to inconsistencies because it is unlikely that a single solution exists that satisfies all the equations exactly. In such cases, a least squares approach is employed to obtain an approximate solution that minimizes the discrepancy between the left and right sides of the equations.

Matrix Representation

Expressing a system of linear equations in matrix form, as in Ax = b, is fundamental because it allows the application of linear algebra techniques to analyze and solve the system. This representation simplifies the notation and enables the use of powerful tools such as matrix factorizations, normal equations, and the computation of pseudoinverses to determine least squares solutions.

Normal Equations

Normal equations are derived by setting the derivative of the sum of squared residuals with respect to the unknown vector to zero, leading to the equation A^T A x = A^T b. Solving the normal equations provides the least squares solution. This method relies on the properties of the transpose and the fact that the product A^T A is symmetric and, under suitable conditions, invertible.

Pseudoinverse

The pseudoinverse, often referred to as the Moore-Penrose inverse, is a generalization of the matrix inverse that exists for any matrix, even when the matrix is not square or not of full rank. It is particularly useful for computing least squares solutions because it provides a way to obtain the best-fit solution for overdetermined systems by effectively 'inverting' the matrix in a least-squares sense.

Transcript

00:01 Here we have a system of three equations with four variables.

00:03 Because there are four variables and only three equations, we can't actually solve the system.

00:09 But what we can do is we can learn a little bit more about the system.

00:12 So that's what we're going to do.

00:13 The first thing we're going to do is we're going to isolate x and find out what that's going to equal to.

00:18 And we're going to use equation two for that.

00:20 So we're going to say x is equal to minus 3y plus 3 z plus 3w.

00:31 So now that we have x, let's find out a little bit more information about y, z, and w.

00:35 So what we're going to do is we're going to replace x in the first and the third equations with what we found.

00:42 So we'll start with equation one, where we'll have two times minus 3y plus 3z plus 3w.

00:51 And then we'll have plus y plus z plus w is equal to 1.

00:56 Let's simplify this to minus 6y plus 6 z plus 6w plus y plus z plus w is equal to 1.

01:07 Now let's simplify this and combine all of our like terms.

01:11 So we'll end up with minus 5y plus 7 z plus 7w.

01:17 Well, that's equal to 1.

01:21 So now let's use this to find out what y would be equal to.

01:28 So we'll have minus 5y is equal to 1 minus 7 z minus 7w or y is equal to 1 minus 7 z minus 7 w all over negative 5.

01:43 And if you wanted to change this a little bit, we could actually just put that negative sign out in front so that it makes it a little bit easier to work with.

01:58 So now we have x and y.

02:01 We have x in terms of y, z, and w, and we have y in terms of z and w...

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