Recall: the inclusion of \(\RNr\) as the \(j\)-th coordinate axis of \(\RNrSpc{n}\) is given by the function
\[\CoordInclsn{j}\from \RNr\longrightarrow \RNrSpc{n},\quad \CoordInclsnOf{j}{x}\DefEq (0,\dots,0,x,0,\dots ,0)\]Show that \(\CoordInclsn{j}\) is a linear function.
To see why \(\CoordInclsn{j}\from \RNr\to \RNrSpc{m}\) is linear, we check that it commutes with vector addition and scalar multiplication:
| \(\CoordInclsnOf{j}{x+y}\) | \(= \) | \((0,\dots ,0,x+y,0,\dots,0)\) |
| \(\) | \(=\) | \((0,\dots ,0,x,0,\dots ,0) + (0,\dots ,0,y,0,\dots ,0)\) |
| \(\) | \(= \) | \(\CoordInclsnOf{j}{x} + \CoordInclsnOf{j}{y}\) |
| \(\CoordInclsnOf{j}{tx}\) | \(=\) | \((0,\dots ,0,tx,0,\dots ,0)\) |
| \(\) | \(=\) | \(t\cdot (0,\dots ,0,x,0,\dots ,0)\) |
| \(\) | \(=\) | \(t\cdot \inc_j(0,\dots ,0,x,0,\dots ,0)\) |
Find all coordinate inclusions of \(\RNr\) in \(\RNrSpc{3}\), and compute their effect on a number \(u\) in \(\RNr\).
There are three inclusions of \(\RNr\) in \(\RNrSpc{3}\) as a coordinate axis, namely \(\CoordInclsn{1}\), \(\CoordInclsn{2}\), and \(\CoordInclsn{3}\). The effect of these inclusions on \(u\) in \(\RNr\) is given by
| \(\CoordInclsnOf{1}{u}\) | \(=\) | \((u,0,0)\) |
| \(\CoordInclsnOf{2}{u}\) | \(=\) | \((0,u,0)\) |
| \(\CoordInclsnOf{3}{u}\) | \(=\) | \((0,0,u)\) |
Find the effect of the three coordinate inclusion functions of \(\RNr\) in \(\RNrSpc{3}\) on the number \(-1\).
The effect of the coordinate inclusions of \(\RNrSpc{3}\) on \(-1\) in \(\RNr\) is given by
| \(\CoordInclsnOf{1}{-1}\) | \(=\) | \((-1,0,0)\) |
| \(\CoordInclsnOf{2}{-1}\) | \(=\) | \((0,-1,0)\) |
| \(\CoordInclsnOf{3}{-1}\) | \(=\) | \((0,0,-1)\) |