Online quiz (due Wednesday 24th May, 11:59pm):

Answer all five questions. Each question is worth 2 marks, the total is 10 marks.

• – IMAGE OF A POINT 1

  Find the image of $({\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{3}}{2}})$ under the rotation $R_{\pi /6}$ through an anticlockwise angle of $\frac{\pi }{6}$ in ${ℝ}^2$.
  

  • $(1,0)$,

  • $(0,1)$,

  • $(1,1)$,

  • $({\displaystyle \frac{1}{2}},-{\displaystyle \frac{\sqrt{3}}{2}})$,

  • $({\displaystyle \frac{\sqrt{3}}{2}},{\displaystyle \frac{1}{2}})$.

• – HOW MANY ROTATIONS

  How many of the following 5 linear transformations are equal to a rotation $R_{\theta }$ around the origin?

  
  

  • 0

  • 1

  • 2

  • 3

  • 4

• – HOW MANY LINEAR

  How many of the following five maps are linear transformations?

  $$T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 2x\hfill \\ \hfill x{e}^2+y\hfill \end{array}\right),T\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),T(x,y)=x+y,$$

  
  

  • 0

  • 1

  • 2

  • 3

  • 4

• – WHICH MATRIX

  Let $T$ and $S$ be linear transformations defined by $T(x,y)=(2x-3y,-y)$ and $S(x,y)=(x-y,x)$. What is the matrix corresponding to the linear transformation $T\circ S$?
  

  • ,

  • ,

  • ,

  • ,

  • .

• – IMAGE OF A POINT 2

  Find the image of $(1,0)$ under the composition of linear transformations $\phi R_{\frac{\pi }{4}}{H}_{\frac{\pi }{4}}$, where the matrix of $\phi$ is .
  

  • $(\frac{3}{\sqrt{2}},\frac{7}{\sqrt{2}})$,

  • $(\frac{1}{\sqrt{2}},\frac{3}{\sqrt{2}})$,

  • $(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$,

  • $(-\frac{1}{\sqrt{2}},\frac{3}{\sqrt{2}})$,

  • $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$.

• – SYSTEM OF LINEAR EQUATIONS

  Find the value for $x_1$, $x_2$, $x_3$, $x_4$ and $x_5$ that satisfies these linear equations:

  • $(x_1,x_2,x_3,x_4,x_5)=(0.5,1,-0.5,0,1)$.

  • $(x_1,x_2,x_3,x_4,x_5)=(-0.5,1,-0.5,0,1)$.

  • $(x_1,x_2,x_3,x_4,x_5)=(0.5,1,0.5,0,1)$.

  • $(x_1,x_2,x_3,x_4,x_5)=(0.5,1,-0.5,0,-1)$.

  • $(x_1,x_2,x_3,x_4,x_5)=(0.5,-1,0.5,0,1)$.