row and column vectors

A column vector with \[m\] elements is, as the name implies, a \[m\times1\] matrix, i.e. \[\mathbf{x}=\begin{pmatrix} x_{1}\\x_{2}\\\vdots\\x_{m} \end{pmatrix}\]. Similarly, a row vector is a \[1\times n\] matrix, i.e. \[\mathbf{y}=\begin{pmatrix} y_{1}&y_{2}&\cdots&y_{n} \end{pmatrix}\]. Vectors are conventionally treated as column vectors, a plausible explanation is that if \[\mathbf{x}\] is a vector and \[\mathbf{M}\] is a matrix representing a linear transformation, the product \[\mathbf{M}\mathbf{v}\] is another column vector (i.e. the image of \[\mathbf{v}\] under that transformation).

A property of row and column vectors is that \[\mathbf{x}^{T}=\mathbf{y}\] and vice versa, \[\mathbf{y}^{T}=\mathbf{x}\]. Taking advantage of this, sometimes column vectors are written as \[\mathbf{x}=\begin{pmatrix} x_{1}&x_{2}&\cdots&x_{m} \end{pmatrix}^{T}\].