Page 98 - Modul Aljabar

b.u = (–3, 2, 1) dan v = (2, 1, 0)
<u, v> = <(–3, 2, 1), (2, 1, 0)>

= 3(–3).2 + 5.2.1 – 1.0
= – 18 + 10 – 0 = – 8

3. Misal u, v ∈ R2 dengan u = (x1, y1) dan v = (x2, y2).

Tentukan apakah <u, v>
berikut merupakan hasil dalam di R2!

a. <u, v> = 3x1x2 + 5y1y2
b. <u, v> = x1x2 – 2y1y2

Penyelesaian :

a. Misal u = (x1, y1), v = (x2, y2) dan w = (x3, y3)
Buktikan aksioma 1

<u, v> = <v, u>
<u, v> = <(x1, y1), (x2, y2)>

= 3x1x2 + 5y1y2

= 3x2x1 + 5y2y1
= <(x2, y2), (x1, y1)>

= <u, v>
Memenuhi aksioma 1

Buktikan aksioma 2
<u+v, w> = <u, w> + <v, w>

<u+v, w> = <(x1, y1) + (x2, y2), (x3, y3)>

= <(x1 + x2, y1 + y2), (x3, y3)>
= 3(x1 + x2 ).x3 + 5(y1 + y2).y3

= (3x1 + 3x2).x3 + (5y1 + 5y2).y3
= 3 x1x3 + 3 x2x3 + 5 y1y3 + 5 y2y3

= (3 x1x3 + 5 y1y3) + (3 x2x3 + 5 y2y3)

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