Problems

Exercise 1.

Give geometric interpretation of kernel and image of a linear operator \(\,F\ \) given by

\[F(\vec{r})\,:\,=\,\vec{a}\times\vec{r}\,,\qquad\vec{r}\in V\,,\]

where \(\,\vec{a}\ \) is a fixed vector from 3-dimensional vector space \(\,V\,\) of geometric vectors.

Describe defect and rank of this operator. \(\,\) Does \(\,\text{def}\,F + \text{rk}\,F = \dim V\ ?\)

Exercise 2.

In 3-dimensional vector space \(\,V\,\) of geometric vectors with basis \(\ \mathcal{E}=(\vec{e}_1,\,\vec{e}_2,\,\vec{e}_3)\ \) consisting of three mutually perpendicular unit vectors, \(\,\) we define a mapping

\[F(\vec{r})\,:\,=\,(\vec{b}\cdot\vec{r})\ \vec{a}\,, \qquad\vec{a},\vec{b},\vec{r}\in V,\quad\vec{a},\vec{b}\ -\ \text{fixed non-zero vectors}.\]

0. Justify that \(\,F\ \) is a linear operator.

1. What is geometric interpretation of kernel and image of the operator \(\,F\,?\)

2. Describe defect and rank of this operator and check the condition \(\,\text{def}\,F + \text{rk}\,F = \dim V.\)

3. Find the matrix \(\,M(F)\ \) of the operator \(\,F\,\) in basis \(\,\mathcal{E}.\)

4. Calculate rank of the matrix \(\,M(F)\ \) and \(\,\) check that \(\,\text{rk}\,F = \text{rk}\,M(F).\)

Exercise 3.

Let \(\,F\in\text{Hom}(V,W),\ \) where \(\,V\ \) and \(\ W\ \) are finite dimensional vector spaces over a field \(\,K.\ \) Verify correctness of the following statements for vectors \(\,v_1,\,v_2,\,\dots,\,v_r\in V\ \) (l.i. = linearly independent, \(\,\) l.d. = linearly dependent):

1. \(\quad v_1,\,v_2,\,\dots,\,v_r\ \ -\ \ \text{l.i.} \qquad\Rightarrow\qquad Fv_1,\,Fv_2,\,\dots,\,Fv_r\ \ -\ \ \text{l.i.}\)

2. \(\quad v_1,\,v_2,\,\dots,\,v_r\ \ -\ \ \text{l.d.} \qquad\Rightarrow\qquad Fv_1,\,Fv_2,\,\dots,\,Fv_r\ \ -\ \ \text{l.d.}\)

3. \(\quad Fv_1,\,Fv_2,\,\dots,\,Fv_r\ \ -\ \ \text{l.i.} \qquad\Rightarrow\qquad v_1,\,v_2,\,\dots,\,v_r\ \ -\ \ \text{l.i.}\)

4. \(\quad Fv_1,\,Fv_2,\,\dots,\,Fv_r\ \ -\ \ \text{l.d.} \qquad\Rightarrow\qquad v_1,\,v_2,\,\dots,\,v_r\ \ -\ \ \text{l.d.}\)