# Why decision boundary is of (D-1) dimensions?

I am reading Bishop - Pattern Recognition and Machine Learning book. In Chapter 4 first page, there is a statement that

In this chapter, we consider linear models for classification, by which we mean that the decision surfaces are linear functions of the input vector x and hence are defined by (D − 1)-dimensional hyper planes within the D-dimensional input space.

If the input vector is of D dimensions then the decision boundary should also be of D dimensions. Why is it D-1? For example, in 2-D plane, the line separating the points is also a straight line which is of 2 dimensions.

Am I thinking something terribly wrong? Can anyone explain me how is it D-1 dimensions for decision boundary?

• Not so; the line is considered 1 dimensional. – Nick Cox Jan 8 '18 at 13:49
• There is a unique $D$-dimensional linear (or affine) subspace of any (finite) $D$-dimensional space: the space itself. That would be useless as a decision boundary, wouldn't it? – whuber Jan 8 '18 at 14:02

In the images below the decision boundaries for 2D and 3D input space are shown in blue. The orange lines illustrate that these decision boundaries have D$-1$ dimensions themselves.
• I agree that a straight line is a 1D object and a plane is a 2D object. But if the input space has two co-ordinates or features then we represent the straight line or decision boundary as ax+by+c=0 i.e., by using two dimensions. Is the author in this context discussing only about their physical appearance or their representation in vector form? – bharadwaj Jan 9 '18 at 15:13
• @bharadwaj $ax+by+c=0$ is because it exists in 2D space, but it is itself a 1D slice of that 2D space. In my images, the number of black lines (i.e. the axes) show the dimensionality of the space but the orange lines show the dimensionality of the boundary and it is always one less than that of the space. – Dan Jan 9 '18 at 15:26