Notes
- Consider [latex]F\,\mathbf{x}=\left(\begin{array}{c}1 & 0\\0 & 0\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}x\\0\end{array}\right)[/latex]. So, matrix F here represents a projection onto the x-axis (since the x-coordinate is unchanged, but the y-coordinate changes to 0).
- Consider [latex]G\,\mathbf{x}=\left(\begin{array}{c}0 & 0\\0 & 1\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}0\\y\end{array}\right)[/latex]. So, matrix G here represents a projection onto the y-axis (since the x-coordinate changes to 0, but the y-coordinate is unchanged).
- Consider a projection onto the line through the origin with direction vector [latex]\mathbf{d}=\left(\begin{array}{c}d_1\\d_2\end{array}\right)[/latex]. We can derive the matrix for this transformation by considering the effects on the standard unit vectors, [latex]\mathbf{e}_1=\left(\begin{array}{c}1\\0\end{array}\right)[/latex] and [latex]\mathbf{e}_2=\left(\begin{array}{c}0\\1\end{array}\right)[/latex]. The projection of [latex]\mathbf{e}_1[/latex] onto the line is [latex]\text{Proj}_\mathbf{d}(\mathbf{e}_1)=\left(\frac{\mathbf{d}\cdot\mathbf{e}_1}{\mathbf{d}\cdot\mathbf{d}}\right)\mathbf{d}=\frac{d_1}{d_1^2+d_2^2}\left(\begin{array}{c}d_1\\d_2\end{array}\right)[/latex]. Similarly, the projection of [latex]\mathbf{e}_2[/latex] onto the line is [latex]\text{Proj}_\mathbf{d}(\mathbf{e}_2)=\left(\frac{\mathbf{d}\cdot\mathbf{e}_2}{\mathbf{d}\cdot\mathbf{d}}\right)\mathbf{d}=\frac{d_2}{d_1^2+d_2^2}\left(\begin{array}{c}d_1\\d_2\end{array}\right)[/latex].
- Therefore the matrix H that represents a projection onto the line through the origin with direction vector [latex]\mathbf{d}=\left(\begin{array}{c}d_1\\d_2\end{array}\right)[/latex] is [latex]H=\frac{1}{d_1^2+d_2^2}\left(\begin{array}{c}d_1^2 & d_1d_2\\ d_1d_2 & d_2^2\end{array}\right)[/latex].
- For example, the matrix that represents a projection onto the line y = 3x, which is the line through the origin with direction vector [latex]\mathbf{d}=\left(\begin{array}{c}1\\3\end{array}\right)[/latex] is [latex]H=\frac{1}{10}\left(\begin{array}{c}1 & 3\\ 3 & 9\end{array}\right)[/latex].
Video Tips
Practice Exercises
Drag and drop the matrices below to their correct locations. Some matrices don’t fit anywhere.
