When multiplying a scalar by a vector (or matrix), we multiply each element of the vector (or matrix) by that scalar. For \(\mathbf{x}_{n}=\left[x_{n 1}, x_{n 2}\right]^{\top}, \mathbf{t}=\left[t_{1}, \ldots, t_{N}\right]^{\top}\), \(\mathbf{w}=\left[w_{0}, w_{1}\right]^{\top}\) and

\[\mathbf{x}=\left[\begin{array}{c} \mathbf{x}_{1}^{\top} \\ \mathbf{x}_{2}^{\top} \\ \vdots \\ \mathbf{x}_{N}^{\top} \end{array}\right]\]

show that

\[\sum_{n} \mathbf{x}_{n} t_{n}=\mathbf{X}^{\top} \mathbf{t}\]

and

\[\sum_{n} \mathbf{x}_{n} \mathbf{x}_{n}^{\top} \mathbf{w}=\mathbf{X}^{\top} \mathbf{X} \mathbf{w}\]