Some test problems
Here are some problems to test your basic knowledge and understanding. Problem 1 The absolute value \(|x|\) of a real number \(x\) is defined as \[|x| = \left\{ \begin{array}{cc} x & \text{if $x \geq 0$},\\ -x & \text{if $x < 0$}. \end{array}\right.\] Prove the following. For any \(x \in \mathbb{R}\), \(-|x| \leq x \leq |x|\). Let \(a > 0\). For any \(x\in\mathbb{R}\), if \(-a \leq x \leq a\), then \(|x| \leq a.\) [5 marks each; 10 marks in total] Problem 2 Let \(\mathbf{a} = (2,3)\), \(\mathbf{b} = (1, 5)\), and \(\mathbf{r} = (x,y)\) be position vectors. The equation \[\mathbf{r} = s\mathbf{a} + (1 - s)\mathbf{b}\] with \(0 \leq s \leq 1\) defines the line segment between \(\mathbf{a}\) and \(\mathbf{b}\). Draw the line segment defined above in the \(x\)-\(y\) plane. Find the closest point on this line segment from the origin \(\mathbf{o} = (0,0)\). [5 marks each; 10 marks in total] Problem 3 Compute the matrix determinant ...