Define the function $\mathrm{\delta }\left(i\,j\right)\=\{\begin{array}{cc}1& i\=j\\ 0& i\ne j\end{array}$ 

$\mathrm{\δ}\left(i\,j\right)\=\left\{\begin{array}{cc}1& i\=j\\ 0& i\ne j\end{array}\right.$$\stackrel{\text{assign as function}}{\to }$$\mathrm{\delta }$

Define the vectors ${\mathbf{U}}_k$ and ${\mathbf{V}}_k$, $k\=1\,2$ 

$\mathbf{U}\=\left[⟨2\,-3⟩\,⟨3\,4⟩\right]$$\stackrel{\text{assign}}{\to }$

$\mathbf{V}\=\left[⟨a_1\,a_2⟩\,⟨b_1\,b_2⟩\right]$$\stackrel{\text{assign}}{\to }$

Form the set of equations  ${\mathbf{U}}_i·{\mathbf{V}}_j\={\mathrm{\δ}}_{\mathrm{ij}}$

${\mathbf{U}}_i·{\mathbf{V}}_j\=\mathrm{\δ}\left(i\,j\right)$

${\left[\left[\begin{array}{c}2\\ \mathrm{-3}\end{array}\right]\,\left[\begin{array}{c}3\\ 4\end{array}\right]\right]}_i·{\left[\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\,\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\right]}_j=\left\{\begin{array}{cc}1& i=j\\ 0& i\ne j\end{array}\right.$

$\stackrel{\text{sequence w.r.t. i}}{\to }$

$\left[\begin{array}{c}2\\ \mathrm{-3}\end{array}\right]·{\left[\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\,\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\right]}_j=\left\{\begin{array}{cc}1& 1=j\\ 0& 1\ne j\end{array}\right.,\left[\begin{array}{c}3\\ 4\end{array}\right]·{\left[\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\,\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\right]}_j=\left\{\begin{array}{cc}1& 2=j\\ 0& 2\ne j\end{array}\right.$

$\stackrel{\text{to list}}{\to }$

$\left[\left[\begin{array}{c}2\\ \mathrm{-3}\end{array}\right]·{\left[\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\,\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\right]}_j=\left\{\begin{array}{cc}1& 1=j\\ 0& 1\ne j\end{array}\right.\,\left[\begin{array}{c}3\\ 4\end{array}\right]·{\left[\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\,\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]\right]}_j=\left\{\begin{array}{cc}1& 2=j\\ 0& 2\ne j\end{array}\right.\right]$

$\stackrel{\text{sequence w.r.t. j}}{\to }$

$\left[2a_1-3a_2=1\,3a_1+4a_2=0\right],\left[2b_1-3b_2=0\,3b_1+4b_2=1\right]$

$\stackrel{\text{list join}}{\to }$

$\left[2a_1-3a_2=1\,3a_1+4a_2=0\,2b_1-3b_2=0\,3b_1+4b_2=1\right]$

$\stackrel{\text{solve}}{\to }$

$\left\{a_1=\frac{4}{17}\,a_2=-\frac{3}{17}\,b_1=\frac{3}{17}\,b_2=\frac{2}{17}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$S$

Evaluate each $V_k$ at the solution $S$ 

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathrm{\delta }\left(i\,j\right)\=\{\begin{array}{cc}1& i\=j\\ 0& i\ne j\end{array}$ 

$\mathrm{\δ}≔\left(i\,j\right)\to \mathrm{piecewise}\left(i\=j\,1\,i\ne j\,0\right)\:$

$U≔\left[⟨2\,-3⟩\,⟨3\,4⟩\right]\:$

$V≔\left[⟨a_1\,a_2⟩\,⟨b_1\,b_2⟩\right]\:$

$q≔\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{DotProduct}\left({\mathbf{U}}_i\,{\mathbf{V}}_j\right)\=\mathrm{\delta }\left(i\,j\right)\,i\=1..2\right)\,j\=1..2\right)$

$q≔2a_1-3a_2=1,3a_1+4a_2=0,2b_1-3b_2=0,3b_1+4b_2=1$

$S≔\mathrm{solve}\left(\left\{q\right\}\right)$

$S≔\left\{a_1=\frac{4}{17}\,a_2=-\frac{3}{17}\,b_1=\frac{3}{17}\,b_2=\frac{2}{17}\right\}$