Some test problems

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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.

  1. For any \(x \in \mathbb{R}\), \(-|x| \leq x \leq |x|\).
  2. 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 \(\mathbf{b}\). 
  1. Draw the line segment defined above in the \(x\)-\(y\) plane.
  2. Find the closest point on this line segment from the origin \(\mathbf{o} = (0,0)\). 
[5 marks each; 10 marks in total]

Problem 3

  1. Compute the matrix determinant \[\begin{vmatrix} a_{11} & a_{12}\\ 0 & a_{22} \end{vmatrix}\] [5 marks]
  2. Compute the matrix determinant \[\begin{vmatrix} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 & a_{33} \end{vmatrix}\] [5 marks]
  3. "Extrapolate" the value of the matrix determinant \[\begin{vmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n}\\ 0 & a_{22} & a_{23} & \cdots & a_{2n}\\ 0 & 0 & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_{nn} \end{vmatrix}\] where the matrix elements are such that \(a_{ij} = 0\) if \(i > j\). Such a matrix is called an upper triangular matrix. [5 marks]
  4. Prove the result of Part 3 by mathematical induction. [10 marks]
[25 marks in total]

Problem 4

Let \(\mathbf{u} = (a, b, 0), \mathbf{v} = (c, d, 0)\in\mathbb{R}^3\).
  1. Compute the scalar product \(\braket{\mathbf{u}, \mathbf{v}}\).
  2. Compute the vector product \(\mathbf{u}\times\mathbf{v}\).
  3. Find the (signed) area of the parallelogram defined by the origin \(\mathbf{0} = (0, 0, 0)\), \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{u} + \mathbf{v}\).
[5 marks each; 15 marks in total]

Problem 5

Let \(u = a+ib, v = c + id\), \(w = u + v\) \(\in \mathbb{C}\) where \(a, b, c, d \in\mathbb{R}\), and \(i\) is the imaginary unit.
  1. Find \(\bar{u}v\). (\(\bar{u}\) means the complex conjugate of \(u\).)
  2. On the complex plane, the four points \(0\), \(u\), \(v\) and \(w\) comprise a parallelogram. Show that its (signed) area is given by \(\Im (\bar{u}v)\), the imaginary part of \(\bar{u}v\).
[5 marks each; 10 marks in total]

Problem 6

Let the set \(\mathbf{C}\) be defined by
\[\mathbf{C} = \left\{ \begin{pmatrix} a & -b\\ b & a \end{pmatrix} \mid a, b \in \mathbb{R}\right\}.\]
  1. \(\mathbf{C}\) is closed under matrix addition. [5 marks]
  2. \(\mathbf{C}\) is closed under matrix multiplication. [5 marks]
  3. Matrix multiplication in \(\mathbf{C}\) is commutative. [5 marks]
  4. Find the inverse matrix of \(\begin{pmatrix}a & -b\\ b & a\end{pmatrix}\) given \(a^2 + b^2 \neq 0\). [5 marks]
  5. \(\mathbf{C}\), with matrix addition and multiplication, is a field. [10 marks]
[30 marks in total]

Answers

See the video:



(These problems were given as the mid-semester test for SM-1201 at UBD.)

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