### Dot product in matrix notation

Given the rules of matrix multiplication, we cannot multiply two vectors when they are both viewed as column matrices. If we try to multiply an $n \times 1$ matrix with another $n \times 1$ matrix, this product is not defined. The number of columns of the first matrix (1) does not match the number of rows of the second matrix ($n$). To rectify this problem, we can take the transpose of the first vector, turning it into a $1 \times n$ row matrix. With this change, the product is well defined; the product of a $1 \times n$ matrix with an $n \times 1$ matrix is a $1 \times 1$ matrix, i.e., a scalar.

If we multiply $\vc{x}^T$ (a $1 \times n$ matrix) with any $n$-dimensional vector $\vc{y}$ (viewed as an $n \times 1$ matrix), we end up with a matrix multiplication equivalent to the familiar dot product of $\vc{x} \cdot \vc{y}$: \begin{align*} \vc{x}^T \vc{y} = \left[ \begin{array}{ccccc} x_1& x_2& x_3& \cdots& x_n \end{array} \right] \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right] =x_1y_1+x_2y_2+x_3y_3 + \ldots + x_ny_n = \vc{x} \cdot \vc{y}. \end{align*} Although we won't typically write a dot product as $\vc{x}^T \vc{y}$, you may see it elsewhere. Moreover, you can view this dot product as forming the building block for the general matrix multiplication.

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