Kenneth Kuttler

9.2. CONSTRUCTING THE MATRIX OF A LINEAR TRANSFORMATION 191

Its inverse is − 1√

(a2+b2)b 1√

(a2+b2)a 0

− c√(a2+b2)

a − c√(a2+b2)

b√

(a2 +b2)

a b c

Therefore, the matrix which does the rotating is

− b√a2+b2

− c√(a2+b2)

a a

a√a2+b2

− c√(a2+b2)

b b

0 a2+b2√a2+b2

 cosθ −sinθ 0

sinθ cosθ 00 0 1

 ·− 1√

(a2+b2)b 1√

(a2+b2)a 0

− c√(a2+b2)

a − c√(a2+b2)

b√

(a2 +b2)

a b c

This yields a matrix whose columns are

b2 cosθ+c2a2 cosθ+a4+a2b2

a2+b2

−bacosθ+cb2 sinθ+ca2 sinθ+c2abcosθ+ba3+b3aa2+b2

−(sinθ)b− (cosθ)ca+ ca

 ,

−bacosθ−ca2 sinθ−cb2 sinθ+c2abcosθ+ba3+b3a

a2+b2

a2 cosθ+c2b2 cosθ+a2b2+b4

a2+b2

(sinθ)a− (cosθ)cb+ cb

 ,

 (sinθ)b− (cosθ)ca+ ca−(sinθ)a− (cosθ)cb+ cb(

a2 +b2)

cosθ + c2

Using the assumption that u is a unit vector so that a2 + b2 + c2 = 1, it follows the

desired matrix is



cosθ −a2 cosθ

+a2−bacosθ +ba−csinθ

(sinθ)b− (cosθ)ca+ca

−bacosθ +ba+csinθ

−b2 cosθ +b2

+cosθ

−(sinθ)a− (cosθ)cb+cb

−(sinθ)b− (cosθ)ca+ca

(sinθ)a− (cosθ)cb+cb

(1− c2

)cosθ

+c2

This was done under the assumption that |c| ≠ 1. However, if this condition does not

hold, you can verify directly that the above still gives the correct answer.

9.2, CONSTRUCTING THE MATRIX OF A LINEAR TRANSFORMATIONIts inverse is— | b | a 0(+0) V(@+0)Teeny" — Tew” Jae +B?)a b cTherefore, the matrix which does the rotating is— 2 > “<q aVerb V(@+8?) cos@ —sin@ 0a Cc .Vai Vesey” b sin@ cos@ 0O0 ee, 0 oO 1Vae+b?—____—p 1___q 0(+0) (+0)_ c _ cp 24 pp(ere) (exe) (a° +b*)a b cThis yields a matrix whose columns areb? cos 0+c2a? cos 0+a4+a7b?a+b—bacos 0+cb? sin +ca* sin 0+c2abcos 0+ba3+b3.aa?+b2 ’— (sin @)b— (cos @)ca+ca—bacos @—ca? sin @—cb* sin 0+c2abcos 0+ba? +b.aa2+b2@ cos 0-+c7b? cos 0+a2b? +b*a+b? ,(sin @)a—(cos0@)cb+cb(sin @)b—(cos@)ca+ca— (sin @)a—(cos0@)cb+cb(a? +b’) cos@ +c?191Using the assumption that u is a unit vector so that a* + b* +c? = 1, it follows thedesired matrix iscos 0 —a’ cos @ —bacos @ + ba (sin 8) b — (cos 8) ca+a? —csin@ +ca—bacos 6 + ba —b* cos 0 +b? — (sin @)a—(cos@) cb+csin @ +cos 6 +cb—(sin@)b—(cos@)ca —_ (sin@)a—(cos@) cb (1—c) cos@+ca +cb +c?This was done under the assumption that |c| 4 1. However, if this condition does nothold, you can verify directly that the above still gives the correct answer.