I am trying to self-study Kevin Murphy's book on machine learning. i am trying to be 100% sure that i understand the notation in the book. . I am however struggling to understand what is meant with the line following "similarly, if we…".
I read this as a column vector, running verticallly with index j from 1:D (dimensions) and each entry itself being a vector (as the x's are bolded) running with index i from 1:N. However – i am confused – as would this not be a matrix? i am surely not understanding something here – it would be great if someone could be so kind to give an example / visual on this would look in practice.
Best Answer
My understanding is that he means the following. The column vector $\mathbf{x_{1}} = \left[ \begin{array}{c} 1\\ 2\\ 3 \end{array} \right]$ would be written $\mathbf{x_{1}} = \left[ 1, 2, 3 \right]$ and the column vector $\mathbf{x_{2}} = \left[ \begin{array}{c} 4\\ 5\\ 6 \end{array} \right]$ would be written as $\mathbf{x_{2}} = \left[ 4, 5, 6 \right]$ then
$\mathbf{a} = [\mathbf{x_{1}}, \mathbf{x_{2}}] = \left[ \begin{array}{c} 1\\ 2\\ 3\\ 4\\ 5\\ 6 \end{array} \right]$ and
$\mathbf{A} = [\mathbf{x_{1}}, \mathbf{x_{2}}] = \left[ \begin{array}{c} 1 & 4\\ 2 & 5\\ 3 & 6\\ \end{array} \right]$