Exercise · ZW-LINEAR-LEAST-SQUARES-007 · ★★☆ Intermediate
With \(A^\top A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\) and \(A^\top b=(3,3)^\top\), what is the least squares solution?
Choose one option.
Show answer
- A. \(x=(1,1)^\top\)
Explanation
Solving \(2x_1+x_2=3\) and \(x_1+2x_2=3\) gives \(x=(1,1)^\top\). The option \((3,3)^\top\) takes \(A^\top b\) directly as the solution without solving the system.
Review the concept: Linear least squares
Optimization problem that minimizes the sum of squared residuals \(\|Ax-b\|_2^2\) to approximate a vector \(b\) by linear combinations of the columns of \(A\).
\[\min_{x}\ \|Ax-b\|_2^2 \quad \Rightarrow \quad A^\top A\,x = A^\top b\]
More practice problems: linear least squares
- ZW-LINEAR-LEAST-SQUARES-001 What is the gradient of f(x)=\|Ax-b\|_2^2 with respect to x?
- ZW-LINEAR-LEAST-SQUARES-002 What are the normal equations of the least squares problem?
- ZW-LINEAR-LEAST-SQUARES-003 If A has linearly independent columns, how many solutions does the least squares problem have?
- ZW-LINEAR-LEAST-SQUARES-004 What does Ax^* represent geometrically, where x^* is the least squares solution?
- ZW-LINEAR-LEAST-SQUARES-005 If A is a 3× 2 matrix, what is the dimension of A^top A?
- ZW-LINEAR-LEAST-SQUARES-006 Given A=[1 0; 0 1; 1 1] and b=(1,2,4)^top, what is A^top b?
- ZW-LINEAR-LEAST-SQUARES-008 To fit y=c t (no intercept) to the points (1,2) and (2,3), what is c?
- ZW-LINEAR-LEAST-SQUARES-009 Why is it not advisable to compute the least squares solution as x=(A^top A)^-1A^top b when A^top A…