A First Course in Linear Algebra, 2017a

166 R n
Notice that this proof was based only on the properties of the dot product listed in Proposition 4.27.
This means that whenever an operation satisfies these properties, the Cauchy Schwarz inequality holds.
There are many other instances of these properties besides vectors in R n .
The Cauchy Schwarz inequality provides another proof of the triangle inequality for distances in R n .
Theorem 4.30: Triangle Inequality
For ⃗u,⃗v ∈ R n ‖⃗u +⃗v‖≤‖⃗u‖ + ‖⃗v‖ (4.9)
and equality holds if and only if one of the vectors is a non-negative scalar multiple of the other.
Also
|‖⃗u‖−‖⃗v‖| ≤ ‖⃗u −⃗v‖ (4.10)
Proof. By properties of the dot product and the Cauchy Schwarz inequality,
Hence,
‖⃗u +⃗v‖ 2 = (⃗u +⃗v) • (⃗u +⃗v)
= (⃗u •⃗u)+(⃗u •⃗v)+(⃗v •⃗u)+(⃗v •⃗v)
= ‖⃗u‖ 2 + 2(⃗u •⃗v)+‖⃗v‖ 2
≤ ‖⃗u‖ 2 + 2|⃗u •⃗v| + ‖⃗v‖ 2
≤ ‖⃗u‖ 2 + 2‖⃗u‖‖⃗v‖ + ‖⃗v‖ 2 =(‖⃗u‖ + ‖⃗v‖) 2
‖⃗u +⃗v‖ 2 ≤ (‖⃗u‖ + ‖⃗v‖) 2
Taking square roots of both sides you obtain 4.9.
It remains to consider when equality occurs. Suppose ⃗u =⃗0. Then, ⃗u = 0⃗v and the claim about when
equality occurs is verified. The same argument holds if ⃗v =⃗0. Therefore, it can be assumed both vectors
are nonzero. To get equality in 4.9 above, Theorem 4.29 implies one of the vectors must be a multiple of
the other. Say⃗v = k⃗u. Ifk < 0 then equality cannot occur in 4.9 because in this case
⃗u •⃗v = k‖⃗u‖ 2 < 0 < |k|‖⃗u‖ 2 = |⃗u •⃗v|
Therefore, k ≥ 0.
To get the other form of the triangle inequality write
so
Therefore,
Similarly,
⃗u =⃗u −⃗v +⃗v
‖⃗u‖ = ‖⃗u −⃗v +⃗v‖
≤‖⃗u −⃗v‖ + ‖⃗v‖
‖⃗u‖−‖⃗v‖≤‖⃗u −⃗v‖ (4.11)
‖⃗v‖−‖⃗u‖≤‖⃗v −⃗u‖ = ‖⃗u −⃗v‖ (4.12)
It follows from 4.11 and 4.12 that 4.10 holds. This is because |‖⃗u‖−‖⃗v‖| equals the left side of either 4.11
or 4.12 and either way, |‖⃗u‖−‖⃗v‖| ≤ ‖⃗u −⃗v‖.
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