4.5 Algebraic Properties of Vector Transposes

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 4.4 Transpose of a Vector

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Algebraic Properties of Vector Transposes

Let $\vec{x}, \vec{y} \in R^{n}$ and $a, b \in R$ . Then, all of the following are properties of 4.2 Column Vector Addition.

$({\vec{x}}^{T})^{T} = \vec{x}$
$(\vec{x} + \vec{y})^{T} = {\vec{x}}^{T} + {\vec{y}}^{T}$
$(a \cdot \vec{x})^{T} = a \cdot [{\vec{x}}^{T}]$

\begin{proof}
Let $\vec{x}, \vec{y} \in R^{n}$ and $a \in R$ . We start off by proving our first property. Consider:

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

This shows that $({\vec{x}}^{T})^{T} = \vec{x}$ . Next, let's prove that the transpose of a sum of vectors is equal to the sum of their transposes. Consider:

\begin{aligned} (\vec{x} + \vec{y})^{T} & = {([\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] + [\begin{array}{c} y_{1} \\ y_{2} \\ ⋮ \\ y_{n} \end{array}])}^{T} \\ = {[\begin{array}{c} x_{1} + y_{1} \\ x_{2} + y_{2} \\ ⋮ \\ x_{n} + y_{n} \end{array}]}^{T} \\ = [\begin{array}{c} x_{1} + y_{1} & x_{2} + y_{2} & \dots & x_{n} + y_{n} \end{array}] \\ = [\begin{array}{c} x_{1} & x_{2} & \dots & x_{n} \end{array}] + [\begin{array}{c} y_{1} & y_{2} & \dots & y_{n} \end{array}] \\ = {\vec{x}}^{T} + {\vec{y}}^{T} \end{aligned}

This shows $(\vec{x} + \vec{y})^{T} = {\vec{x}}^{T} + {\vec{y}}^{T}$ . Finally, let's prove scalar multiplication goes through the transpose operation. Consider:

\begin{aligned} (a \cdot \vec{x})^{T} & = {(a \cdot [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}])}^{T} \\ = {[\begin{array}{c} a \cdot x_{1} \\ a \cdot x_{2} \\ ⋮ \\ a \cdot x_{n} \end{array}]}^{T} \\ = [\begin{array}{c} a \cdot x_{1} & a \cdot x_{2} & \dots & a \cdot x_{n} \end{array}] \\ = a \cdot [\begin{array}{c} x_{1} & x_{2} & \dots & x_{n} \end{array}] \\ = a \cdot {\vec{x}}^{T} \end{aligned}

This completes our proof.
\end{proof}
Remark. Although the properties are shown with column vectors, the same holds true if $\vec{x}, \vec{y}$ were row vectors. As we will see later, the transpose operation is extremely helpful in re-interpreting dot products between vectors as matrix-matrix multiplication. Transposes are also helpful in formulating outer-products between vectors.
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