Notes - Linear Algebra I MT22, Triangle Inequality


Flashcards

Can you state the triangle inequality in full for an inner product space?


For $v, w$ in an inner product space $V$, then

\[||v+w|| \le ||v|| + ||w||\]

How would you “expand” $\langle v+w, v + w \rangle$?


\[\langle v, v\rangle + 2\langle v, w\rangle + \langle w, w\rangle\]

In the proof of the triangle inequality for an inner product space, i.e.

For $v, w$ in an inner product space $V$, then $ \vert \vert v+w \vert \vert \le \vert \vert v \vert \vert + \vert \vert w \vert \vert $.

What justifies the last line of working:

\[\begin{aligned} ||v+w||^2 &= \langle v, v\rangle + 2\langle v, w\rangle + \langle w, w\rangle \\\\ &=||v||^2 + 2||v||\text{ }||w|| + ||w||^2 \end{aligned}\]

The Cauchy-Schwarz inequality.




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