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Geometric Linear Algebra
63 SOLVED PROBLEMS
Question: 2.1
In R2, fix the following points O = (1, 0), A1 = (1, 2), A2 = (0, 1), and O′ = (−1,−1), B1 = (0, 0), B2 = (2, 3). Construct the vectorized spaces Σ(O;A1,A2) and Σ(O′;B1,B2), and then use them to justify the content of (2.4.2). ...
Question: 3.4
The linear operator A= [λ 0 0 b λ 0 0 c λ] , where bc ≠ 0 has the following properties: 1. A satisfies its characteristic polynomial −(t − λ)3, i.e. (A − λI3)3 = O3×3 but A − λI3 = [0 0 0 b 0 0 0 c 0] ≠ O3×3, (A − λI3)2 =[0 0 0 0 0 0 bc 0 0] ≠ O3×3. 2. Hence, A has eigenvalues λ, λ, λ with ...
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. ...
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. ...
Question: 3.37
Determine the relative positions of two planes S21 = x0^→ + S1 and S22 = y0^→ + S2 in R3(see Fig. 3.16). ...
Question: 3.36
Determine the relative positions of two lines S11 = x0^→ + S1 and S12 = y0^→ + S2 in R3 (see Fig.3.13). ...
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. ...
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^→ ). ...
Question: 3.26
Analyze the rational canonical form ...
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 ...
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