4.4 Transpose of a Vector

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Properties: 4.5 Algebraic Properties of Vector Transposes
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Transpose of a Vector

Let $\vec{x} \in R$ be a 3.1 Column Vector. We define the transpose of $\vec{x}$ to be the $1 \times n$ 3.4 Row Vector

{\vec{x}}^{T} = [\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \end{matrix}]

where the entry in the $i$ th row of $\vec{x}$ becomes the entry in the $i$ th column of ${\vec{x}}^{T}$ . Think of "turning" the vector on its side.

Let $\vec{y} \in R^{1 \times n}$ be a row vector. The tranpose of $\vec{y}$ is the $n \times 1$ column vector

{\vec{y}}^{T} = [\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{matrix}]

where the coefficient in the $k$ th column of $\vec{y}$ becomes the entry in the $k$ th row of ${\vec{y}}^{T}$ .