If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related?

If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related?

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If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related? How are $uv^T$ and $vu^T$ related?

If $u jeffp 1 Answer Answered by 2 years ago 8.7k points we have$u,v \in \Bbb R^n$with$v=\begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix}$and$v^T=\begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix}u=\begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix}$and$u^T=\begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix}$how are$u^tv$and$v^tu$related $$u^Tv=\begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix} \begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix}=u_1v_1+u_2v_2+\ldots+u_nv_n$$ $$v^Tu=\begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix} \begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix}=u_1v_1+u_2v_2+\ldots+u_nv_n$$ so$u^Tv=v^Tu$How are$uv^T$and$vu^T$related $$uv^T=\begin{bmatrix}u_1 \\ u_2 \\ \vdots \\ u_n\end{bmatrix} \begin{bmatrix}v_1 & v_2 & \ldots & v_n\end{bmatrix} = \begin{bmatrix}u_1v_1 & u_1v_2 & \ldots & u_1v_n \\ u_2v_1 & u_2v_2 & \ldots & u_2v_n \\ \vdots & & \ddots & \\ u_nv_1 & u_nv_2 & & u_nv_n \end{bmatrix}$$ $$vu^T=\begin{bmatrix}v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix} \begin{bmatrix}u_1 & u_2 & \ldots & u_n\end{bmatrix} = \begin{bmatrix}v_1u_1 & v_1u_2 & \ldots & v_1u_n \\ v_2u_1 & v_2u_2 & \ldots & v_2u_n \\ \vdots & & \ddots & \\ v_nu_1 & v_nu_2 & & v_nu_n \end{bmatrix}$$ so$uv^T=(vu^T)^T$Your Answer Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0\$ or $$\beta^2-1=0$$

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