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Exercise 1: Computations of Determinants

Five $3\times 3$-matrices are given.

$$ \begin{matr}{rrr} 2 & -3 & 1 \\\\ -1 & 2 & 2 \\\\ 3 & -5 & 1 \end{matr},\, \,\begin{matr}{rrr} 4 & 2 & 1 \\\\0 & 1 & 2 \\\\ 0 & 0 & 2 \end{matr},\, \, \begin{matr}{rrr} 2 &0 & 0 \\\\ 3 & 5 & 0 \\\\ 9 & -4 & 3 \end{matr},\, \, \begin{matr}{ccc} 2-i & 0 & 0 \\\\ 0 & 2+i & 0 \\\\ 0 & 0 & 1 \end{matr},\, \, \begin{matr}{rrr} 4 & 2 & 1 \\\\1 & 1 & 2 \\\\ 1 & 1 & 2 \end{matr}\,. $$
A

By hand: Compute the determinant of the first matrix by expansion along a row or column of your own choice.

B

By mental calculation: Compute the determinant of the remaining four matrices.

Exercise 2: Parametric Representation of a Parallelogram

In the Figure a plane is drawn that contains a parallelogram $A$.

vektor16.png

A

State a parametric representation of $A$.

Exercise 3: Different Bases in Space

We consider the coordinates of space vectors in different bases, see the figure.

abasis02.png

A

It is apparent from the figure above, that $\mathbf a$, $\mathbf b$ and $\mc$ are linearly independent. A basis m is given by $(\mathbf a, \mathbf b,\mathbf c)$. Determine the coordinate vector $_\mathrm m\mathbf d$.

B

It is also apparent from the figure that $(\mathbf a, \mathbf b,\mathbf d)$ is a basis, let us call this basis for n. Determine the coordinate vector $_\mathrm n\mathbf c$.

%####### begin:question %Indtegn med origo som begyndelsespunkt den vektor $\mathbf u$ som med hensyn til basis m har koordinaterne %

$$ % _\mathrm m\mathbf u= %\begin{matr}{r}2\\\\1\\\\1\end{matr}\,. %$$

%####### end:question

Exercise 4: Geometry in the Plane

We consider three geometric vectors in an ordinary coordinate system in the plane:

vektorer_planen_b.png

C

Determine the length of each vector.

D

Determine the angle between $\,\ma\,$ and $\,\mb\,.$

E

Determine using the determinant method the area that is spanned by $\,\ma\,$ and $\,\mb\,.$

F

Determine the length of the projection vector proj($\mb,\mc$).

G

Determine the projection vector proj($\mb,\mc$).

Exercise 5: Geometry in Space

We shall compute the volume of the parallelepiped that is spanned by three geometric vectors in an ordinary coordinate system in space:

paralleleppidum2.png

The coordinates of the vectors are in the standard base given by

$$_\mathrm e\ma=(1,0,0)\,,\,\,_\mathrm e\mb=(1,2,0)\,,\,\,_\mathrm e\mc=\left(\,\frac 32,\frac 12,\frac 32\,\right)\,.$$
A

Find the volume of the parallelepiped using the formula: the area of the base times the height.

B

Determine det$(\left[\,\mathrm e\ma\,\,\mathrm e\mb\,\,_\mathrm e\mc\,\right])\,.$

Exercise 6: Linear Dependence or Independence

Three space vectors are given. In an ordinary coordinate system the three vectors have the coordinates

$$(3,1,5),\,(2,3,9)\,\,\,\mathrm{and}\,\,\,(-5,3,3)\,.$$
A

Determine using Maple the determinant of the matrix that has the three vectors as columns. Are the three vectors linearly independent?

B

A consequence of a correct answer to the first question is that at least one of the three vectors can be written as a linear combination of the others. Write one of the three vectors as a linear combinations of the other two.

C

State the volume of the parallelepiped that is spanned by the vectors $\,(3,1,5),\,(2,3,9)\,\,\,\mathrm{and}\,\,\,(-5,3,3)\,.$

D

Determine a vector that is perpendicular to both $\,(3,1,5)\,$ and $\,(2,3,9)\,,$ and that together with the two vectors spans a parallelepiped with the volume $187\,.$