سوال ۲۶ کنکور ارشد ریاضی ۱۴۰۱

PASSAGE 3 (excerpt): "If one has several vectors in $C^{n}$ or several matrices in $M_{n}$, what might it mean to say that some are 'small' or that others are 'large'? Under what circumstances might we say that two vectors are 'close together' or 'far apart'? Questions of 'size' and 'proximity' in a two- or three-dimensional real vector space usually refer to Euclidean distance. The Euclidean length of a vector $z \in R^{n}$ is $(z^{T}z)^{1/2} = (\sum z_i^{2})^{1/2}$, and $z$ is said to be 'small' (with respect to this measure) if this nonnegative real number is small. The vectors $x$ and $y$, furthermore, are 'close' if the Euclidean length of the difference $z = x - y$ is a small number. ... One way to answer these questions is to study norms, or measures of size, of matrices and vectors. Norms may be thought of as generalizations of Euclidean length..."

Norms are used to show ----------.