Questions

01

Summation notation

Write out the formula for the operation of a $4\times 4$ matrix on a $4$D vector. (Convert the summation notation to ordinary sums.)

02

Stretching and area

Consider the unit square $[0,1]\times [0,1]$. What is the area of the image of this square under the action of the matrix $\left(\begin{array}{cc}{a}_1& 0\\ 0& a_2\end{array}\right)$?

03

Positive pitch

Verify from the formulas that a positive pitch means an airplane is ascending. Does a positive roll imply an airplane is tilting to the right or to the left?

04

Projection to standard basis

Check that ${\mathrm{proj}}_{{\mathbf{e}}_1}$ sends $(x,y)\mapsto (x,0)$ and ${\mathrm{proj}}_{{\mathbf{e}}_2}$ sends $(x,y)\mapsto (0,y)$. What is the matrix of ${\mathrm{proj}}_{{\mathbf{e}}_i}$ in 3D, for $i=1,2,3$?