Find a least-squares solution of Ax = b

Find a least-squares solution of Ax = b for

Solution:

        The least-squares solution of Ax = b is the vector x that minimizes the norm of the residual vector, defined as r = b - Ax.

        To find the least-squares solution, we can set the derivative of the residual vector norm with respect to x equal to zero and solve for x. This results in the equation:

`A^T(b - Ax) = 0`

`A^T A x = A^T b`

`x = (A^T A)^-1 A^T b`

To calculate this solution, we first need to find the transpose of A, which is:

Next, we need to find the product of `A^T A`, which is:

Next, we need to find the product of `A^T b`, which is:

We then need to find the inverse of `(A^T A)`, which is:

Finally, we can find the least-squares solution by taking the product of `(A^T A)^-1` and `A^T b`:

So the least-squares solution of Ax = b for the given matrix and vector is