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Exercise 1: Computations of Determinants

Five $3\times 3$ matrices are given:

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

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

In your head: Compute the determinant of the remaining four matrices.

Exercise 2: Parametric Representation of a Parallelogram

In the figure a plane contains a parallelogram $A$ .

Provide a parametric representation of $A$ .

Exercise 3: Different Bases in 3D Space

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

It is apparent from the figure above, that $\mathbf a$ , $\mathbf b$ and $\mc$ are linearly independent. A basis $\mathrm m$ is given by $(\mathbf a, \mathbf b,\mathbf c)$ . The endpoint of $\mathbf d$ is halfway along the edge opposite of the edge formed by $\mathbf b$ in the parallelepiped. Determine the coordinate vector $_\mathrm m\mathbf d$ .

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

Exercise 4: Geometry in the Plane

We consider three geometric vectors in a standard coordinate system in the plane:

Compute the length of each vector.

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

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

Compute the length of the projection vector $\mathrm{proj}(\mb,\mc)$ .

Compute the projection vector $\mathrm{proj}(\mb,\mc)$ .

Exercise 5: Geometry in 3D Space

We will compute the volume of the parallelepiped that is spanned by three geometric vectors in a standard coordinate system in 3D space:

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

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

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

Determine

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

Exercise 6: Linear Dependence or Independence

The following three vectors in 3D space are given in the standard coordinate system

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

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

A consequence of the answer to question a) 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.

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

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 $187\,.$