Squeezing the vector space $\mathbb{R}^n$ in the direction of one coordinate axis, i.e. multiplying a certain component $x_i$ of all vectors by a non-zero real number $k$ , is a linear transformation of $\mathbb{R}^n$ .

A concrete example of such squeezing and its results is obtained if we squeeze in $\mathbb{R}^2$ , i.e. in the Euclidean plane formed by all pairs $(x,\,y)$ of real numbers, every $y$ -coordinate by a positive number $k = \frac{b}{a}$ where $a > b > 0$ . We may look how this procedure acts on the circle

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In the picture below, the circle $x^2\!+\!y^2=a^2$ is drawn in red and the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in blue. The angle $t$ is the eccentric anomaly at the point $P$ of the ellipse, which has the parametric presentation $x = a\cos{t}$ , $y = b\sin{t}$ .

Squeezing $\mathbb{R}^3$ in the directions of the $y$ -axis and $z$ -axis one can get from the sphere $$x^2\!+\!y^2\!+\!z^2 = a^2$$ the ellipsoid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1.$$