# Why the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} :

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What is the norm of $\frac{x}{\| x \|}$? Then use the scalar-linearity of the mapping $x\mapsto Ax$.

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For every $x\in K^n$ there is a normalized vector $\frac{x}{\|x\|}$; using the axioms for matrixnorms we have $\|A\frac{x}{\|x\|}\|=\frac{\|Ax\|}{\|x\|}$
Therefore, it is sufficient to only consider the maximum over vectors of norm 1. (As a matrix represents a linear function, there cannot be a $k\in K$, for which $A(kx)>kA(x)$.)