R^3/Standard projections/Example

For ${}\mathbb {R} ^{3}$, equipped with the standard basis, we consider subsets ${}J\subset \{1,2,3\}$ with two elements and the corresponding projections (in the sense of example) onto the coordinate planes. The projections are

$p_{\{1,2\}}\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3},(a,b,c)\longmapsto (a,b,0),$

$p_{\{1,3\}}\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3},(a,b,c)\longmapsto (a,0,c),$

$p_{\{2,3\}}\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3},(a,b,c)\longmapsto (0,b,c).$

The images of some object in ${}\mathbb {R} ^{3}$ under these projections carry names like ground view etc.

For a subset ${}\{j\}\subseteq \{1,2,3\}$ with one element, the projections map to an axis.