Recall: the projection of \(\RNrSpc{m}\) onto the \(i\)-th coordinate is the function
\[\PrjctnOnto{i}\from \RNrSpc{m} \longrightarrow \RNr,\qquad \PrjctnOntoOfCoord{i}{(x_1,\dots ,x_m)}=x_i\]Show directly that \(\PrjctnOnto{i}\) is a linear function.
To see why \(\PrjctnOnto{i}\from \RNrSpc{m}\to \RNr\) is linear, we check that it commutes with vector addition and with scalar multiplication
| \(\PrjctnOntoOf{i}{(x_1,\dots ,x_m)+(y_1,\dots ,y_m)}\) | \(= \) | \(\PrjctnOntoOfCoord{i}{(x_1+y_1,\dots ,x_m+y_m)}\) |
| \(\) | \(= \) | \(x_i+y_i\) |
| \(\) | \(= \) | \(\PrjctnOntoOfCoord{i}{(x_1,\dots ,x_m)} + \PrjctnOntoOfCoord{i}{(y_1,\dots ,y_m)}\) |
| \(\PrjctnOntoOf{i}{t\cdot (x_1,\dots ,x_m)}\) | \(=\) | \(\PrjctnOntoOfCoord{i}{(tx_1,\dots ,tx_m)}\) |
| \(\) | \(=\) | \(tx_i\) |
| \(\) | \(=\) | \(t\cdot \PrjctnOntoOfCoord{i}{(x_1,\dots ,x_m)}\) |
List all coordinate projections of \(\RNrSpc{3}\), and find their effect on a vector \((x,y,z)\).
\(\RNrSpc{3}\) has three projection functions, namely \(\PrjctnOnto{1}\), \(\PrjctnOnto{2}\), and \(\PrjctnOnto{3}\). The effect of these projections on \(\Vect{x}=(x,y,z)\) is given by
| \(\PrjctnOntoOfCoord{1}{(x,y,z)}\) | \(=\) | \(x\) |
| \(\PrjctnOntoOfCoord{2}{(x,y,z)}\) | \(=\) | \(y\) |
| \(\PrjctnOntoOfCoord{3}{(x,y,z)}\) | \(=\) | \(z\) |
Find the effect of the three coordinate projections of \(\RNrSpc{3}\) on the vector \(\Vect{x}=(2,5,7)\).
From the previous problem we know that \(\RNrSpc{3}\) has three coordinate projections, namely
| \(\PrjctnOntoOfCoord{1}{(x,y,z)}\) | \(=\) | \(x\) |
| \(\PrjctnOntoOfCoord{2}{(x,y,z)}\) | \(=\) | \(y\) |
| \(\PrjctnOntoOfCoord{3}{(x,y,z)}\) | \(=\) | \(z\) |
Therefore,
| \(\PrjctnOntoOfCoord{1}{(2,5,7)}\) | \(=\) | \(2\) |
| \(\PrjctnOntoOfCoord{2}{(2,5,7)}\) | \(=\) | \(5\) |
| \(\PrjctnOntoOfCoord{3}{(2,5,7)}\) | \(=\) | \(7\) |