# Question 3.4: The linear operator A= [λ 0 0 b λ 0 0 c λ] , where bc ≠ 0 ha...

The linear operator

$A= \left[\begin{matrix} \lambda & 0 & 0 \\ b & \lambda & 0 \\ 0 & c & \lambda \end{matrix} \right],$    where bc ≠ 0

has the following properties:

1. A satisfies its characteristic polynomial −(t − λ)³, i.e.

$\left(A-\lambda I_{3} \right) ^{3}=O_{3\times 3}$

but

$A-\lambda I_{3} = \left[\begin{matrix} 0& 0 & 0 \\ b & 0 & 0 \\ 0 & c & 0 \end{matrix} \right] \neq O_{3\times 3}, \left(A-\lambda I_{3} \right) ^{2}= \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ bc & 0 & 0 \end{matrix} \right] \neq O_{3\times 3}.$

2. Hence, A has eigenvalues λ, λ, λ with associated eigenvectors $t\overrightarrow{e_{1} },t\in R$ and t ≠ 0 and A is not diagonalizable.

3. Notice that A can be written as the sum of the following linear operators:

$A=\lambda I_{3} + \left(A-\lambda I_{3} \right)$

$= \lambda \left[\begin{matrix} 1& 0 & 0 \\ \frac{b}{\lambda } & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] + \left[\begin{matrix} 0& 0 & 0 \\ 0 & 0 & 0 \\ 0 & c & 0 \end{matrix} \right].$

Note that $\ll \overrightarrow{e_{1} } \gg$  is the only invariant line (subspace). See Fig. 3.35 for some geometric feeling.

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Question: 3.35

## Give an affine transformation T(x^→ ) = x0^→ + x^→ A, where A =[1/3 2/3 2/3 2/3 1/3 -2/3 2/3 -2/3 1/3]. Determine these x0^→ so that each such T is an orthogonal reflection, and the direction and the plane of invariant points. ...

It is obvious that A is orthogonal and det A = −1....
Question: 3.38

## Determine the relative positions of (a) two straight lines in R4,(b) two (two dimensional) planes in R4, and (b) two (two-dimensional) planes in R4, and (c) one (two-dimensional) plane and one (three-dimensional) hyperplane in R4. ...

(a) The answer is like Example 3.36. Why? Prove it...
Question: 3.37

## Determine the relative positions of two planes S21 = x0^→ + S1 and S22 = y0^→ + S2 in R3(see Fig. 3.16). ...

Both $S_{1} and S_{2}$ are two-dime...
Question: 3.36

## Determine the relative positions of two lines S11 = x0^→ + S1 and S12 = y0^→ + S2 in R3 (see Fig.3.13). ...

Remind that both $S_{1}$ and ...
Question: 3.34

## Give an affine transformation T(x^→ ) = x0^→ + x^→A, where A = [-1/3 2/3 -2/3 2/3 2/3 1/3 2/3 -1/3 -2/3].Try to determine these x0^→ so that T is a rotation. In this case, determine the axis and the angle of the rotation, also the rotational plane. ...

The three row (or column) vectors of A are of unit...
Question: 3.33

## Let a0^→ = (1, 1, 0),a1^→ = (2, 0,−1) and a2^→ = (0,−1, 1). Try to construct a shearing with coefficient k ≠ 0 in the direction a1^→ − a0^→ with a0^→ +<>⊥ as the plane of invariant points. Note that (a1^→ − a0^→ )⊥ (a2^→ − a0^→ ). ...

Since \overrightarrow{a_{1} }-\overrightarr...
Question: 3.26

## Analyze the rational canonical form ...

Analysis The characteristic polynomial is  ...
Question: 3.30

## (a) Find the reflection of R3 along the direction v3= (−1, 1,−1) with respect to the plane (2,−2, 3) + <>. (b) Show that T(x ^→) = x0^→ + xA^→, where x0^→ = (0,−2,−4) and A = [1 0 0 0 5/3 4/3 0 -4/3 -5/3] is a reflection. Determine its direction and plane of invariant points ...

(a) In the affine basis B = \left\{\left(2,...
Question: 3.32

## Give an affine transformation T(x ) = x0 + x A, where A =[-1 6 -4 2 4 -5 2 6 -7].Try to determine x0 so that T is a two-way stretch. In this case, determine the line of invariant points and the invariant plane. ...

A has characteristic polynomial det \left(A...
(a) In the affine basis \mathcal{B} = \left...