Page 93 - 35Linear Algebra

4.3 Directions and Magnitudes 93

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Theorem 4.3.2 (Triangle Inequality). For any u, v ∈ R
ku + vk ≤ kuk + kvk.

Proof.

ku + vk 2 = (u + v) (u + v)
= u u + 2u v + v v
2
2
= kuk + kvk + 2 kuk kvk cos θ
2
= (kuk + kvk) + 2 kuk kvk(cos θ − 1)
2
≤ (kuk + kvk) .
That is, the square of the left-hand side of the triangle inequality is ≤ the
square of the right-hand side. Since both the things being squared are posi-
tive, the inequality holds without the square;

ku + vk ≤ kuk + kvk

The triangle inequality is also “self-evident” when examining a sketch of
u, v and u + v.

Example 54 Let
   
1 4
3
2
   
a =   and b =   ,
3 2
   
4 1
so that
2
2
2
a a = b b = 1 + 2 + 3 + 4 = 30
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