The aim of the SVM algorithm is to find a hyperplane in an \( p \)-dimensional space, where \( p \) is the number of features that distinctly classifies the data points.
In a \( p \)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \( p-1 \). As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is a two-dimensional subspace, or stated simply, a plane.
In two dimensions, with the variables \( x_1 \) and \( x_2 \), the hyperplane is defined as $$ b+w_1x_1+w_2x_2=0, $$ where \( b \) is the intercept and \( w_1 \) and \( w_2 \) define the elements of a vector orthogonal to the line \( b+w_1x_1+w_2x_2=0 \). In two dimensions we define the vectors \( \boldsymbol{x} =[x1,x2] \) and \( \boldsymbol{w}=[w1,w2] \). We can then rewrite the above equation as $$ \boldsymbol{w}^T\boldsymbol{x}+b=0. $$