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

$$|x|=\{\begin{array}{cc}x& \text{if }x\ge 0,\\ -x& \text{if }x<0.\end{array}$$

Prove the following.

1. For any $x\in \mathbb{R}$, $-|x|\le x\le |x|$.
2. Let $a>0$. For any $x\in \mathbb{R}$, if $-a\le x\le a$, then $|x|\le a.$ 

[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 $0\le s\le 1$ 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 $$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ 0& a_{22}\end{array}\right|$$ [5 marks]
2. Compute the matrix determinant $$\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ 0& a_{22}& a_{23}\\ 0& 0& a_{33}\end{array}\right|$$ [5 marks]
3. "Extrapolate" the value of the matrix determinant $$\left|\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \cdots & a_{1n}\\ 0& a_{22}& a_{23}& \cdots & a_{2n}\\ 0& 0& a_{33}& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & a_{nn}\end{array}\right|$$ 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 $⟨\mathbf{u},\mathbf{v}⟩$.
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 $\overline{u}v$. ($\overline{u}$ 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 $\mathrm{\Im }(\overline{u}v)$, the imaginary part of $\overline{u}v$.

[5 marks each; 10 marks in total]

Problem 6

Let the set $\mathbf{C}$ be defined by

$$\mathbf{C}=\{\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)\mid a,b\in \mathbb{R}\}.$$

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 $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$ given $a^2+b^2\ne 0$. [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.)