Exercise · ZW-LINEAR-LEAST-SQUARES-006 · ★★☆ Intermediate
Given \(A=\begin{pmatrix}1&0\\0&1\\1&1\end{pmatrix}\) and \(b=(1,2,4)^\top\), what is \(A^\top b\)?
Choose one option.
Show answer
- B. \((5,6)^\top\)
Explanation
The first component is \(1+4=5\) and the second is \(2+4=6\), giving \((5,6)^\top\). The option \((5,4)^\top\) forgets to add the third row's contribution to the second component.
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-007 With A^top A=[2 1; 1 2] and A^top b=(3,3)^top, what is the least squares solution?
- 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…