= Solution
The vector <Euclidean norm> is $\boxed{\|x\|_2=(x^Tx)^{1/2}=(\sum_i x_i^2)^{1/2}}$, the ordinary length from the origin. The <induced matrix norm> is
$$
\boxed{\|A\|_2=\sup_{x\ne0}\frac{\|Ax\|_2}{\|x\|_2}=\max_{\|x\|_2=1}\|Ax\|_2.}
$$
Geometrically this is the largest stretch of a unit vector. A linear map sends the unit sphere to an ellipsoid, possibly degenerate; its largest semiaxis has length $\|A\|_2$. Equivalently the <matrix 2-norm> is the largest <singular value>.
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