Question

We define the vectors

$\overrightarrow{{\mathbf{e}}_{\mathbf{1}}}\mathbf{=}\left[\left.\begin{array}{c}1\\ 0\\ 0\end{array}\right],\overrightarrow{e_2}=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\overrightarrow{e_3}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right.\right]$

in${\mathit{R}}^{\mathbf{3}}$.

a. For role="math" localid="1659342928825" $\mathbf{A}\mathbf{=}\mathbf{\left[}\begin{array}{ccc}\mathbf{a}& \mathbf{b}& \mathbf{c}\\ \mathbf{d}& \mathbf{e}& \mathbf{f}\\ \mathbf{g}& \mathbf{h}& \mathbf{k}\end{array}\mathbf{\right]}$

compute role="math" localid="1659343034980" $\mathit{A}\overrightarrow{{\mathbf{e}}_{\mathbf{1}}}\mathbf{,}\mathit{A}\overrightarrow{{\mathbf{e}}_{\mathbf{2}}}$and role="math" localid="1659343045854" $\mathit{A}\overrightarrow{{\mathbf{e}}_{\mathbf{3}}}$.

b. If B is an role="math" localid="1659343084344" $\mathbf{n}\mathbf{\times }\mathbf{3}$matrix with columns $\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}$and $\overrightarrow{{\mathbf{v}}_{\mathbf{3}}}$, what are role="math" localid="1659343268769" $\mathit{B}\overrightarrow{{\mathbf{e}}_{\mathbf{1}}}\mathbf{,}\mathit{B}\overrightarrow{{\mathbf{e}}_{\mathbf{2}}}$and $\mathit{B}\overrightarrow{{\mathbf{e}}_{\mathbf{3}}}$?

Short answer

a. The systems are$A\overrightarrow{e_1}=\left[\left.\begin{array}{c}a\\ d\\ g\end{array}\right],A{\overrightarrow{e}}_2=\left[\begin{array}{c}b\\ e\\ h\end{array}\right],A\overrightarrow{e_3}=\left[\begin{array}{c}c\\ f\\ k\end{array}\right.\right]$

b. The systems are $B\overrightarrow{e_1}=\left[v_1\right],B\overrightarrow{e_2}=\left[v_2\right],B\overrightarrow{e_3}=\left[v_3\right]$

Step-by-step solution

Consider the matrices.

If is an $n\times m$matrix with row vectors $\overrightarrow{{\omega }_1},......,\overrightarrow{{\omega }_n}$and $\overrightarrow{x}$is a vector in ${\mathrm{ℝ}}^m$then,

$A\overrightarrow{x}=\left[\begin{array}{c}-\overrightarrow{\omega 1}-\\ ⋮\\ -\overrightarrow{\omega n}-\end{array}\right]\overrightarrow{x}=\left[\begin{array}{c}-\overrightarrow{\omega 1}.\overrightarrow{x}-\\ ⋮\\ -\overrightarrow{\omega n}.\overrightarrow{x}-\end{array}\right]$

The matrices are,

$A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$

The matrix B is an $n\times 3$matrix with columns $\overrightarrow{v_1},\overrightarrow{v_2}$and $\overrightarrow{v_3}$, is $B=\left[v_1{v}_2{v}_3\right]$,

Step 2:(a) Perform the multiplication operation.

The systems $A{\overrightarrow{e}}_1,A{\overrightarrow{e}}_2,A{\overrightarrow{e}}_3$are,

$$\begin{gathered} A{\overrightarrow{e}}_1=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\Rightarrow A{\overrightarrow{e}}_1=\left[\begin{array}{c}a.1+b.0+c.0\\ d.1+e0+f.0\\ g.1+h.0+k.0\end{array}\right]\Rightarrow A{\overrightarrow{e}}_1=\left[\begin{array}{c}a\\ d\\ g\end{array}\right] \\ A{\overrightarrow{e}}_2=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\Rightarrow A{\overrightarrow{e}}_2=\left[\begin{array}{c}a.0+b.1+c.0\\ d.0+e1+f.0\\ g.0+h.1+k.0\end{array}\right]\Rightarrow A{\overrightarrow{e}}_2=\left[\begin{array}{c}a\\ d\\ g\end{array}\right] \\ A{\overrightarrow{e}}_3=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& j\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\Rightarrow A{\overrightarrow{e}}_3=\left[\begin{array}{c}a.0+b.0+c.1\\ d.0+e0+f.1\\ g.0+h.0+k.1\end{array}\right]\Rightarrow A{\overrightarrow{e}}_3=\left[\begin{array}{c}c\\ f\\ k\end{array}\right] \end{gathered}$$

(b) Perform the multiplication operation.

The systems $B{\overrightarrow{e}}_1,B{\overrightarrow{e}}_2,B{\overrightarrow{e}}_3$are,

$$\begin{gathered} B{\overrightarrow{e}}_1=\left[v_1{v}_2{v}_3\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\Rightarrow B{\overrightarrow{e}}_1=\left[v_1.1_{}{v}_2.0v_3.0\right]\Rightarrow B{\overrightarrow{e}}_1=\left[v_1\right] \\ B{\overrightarrow{e}}_2=\left[v_1{v}_2{v}_3\right]\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\Rightarrow B{\overrightarrow{e}}_2=\left[v_1.0_{}{v}_2.1v_3.0\right]\Rightarrow B{\overrightarrow{e}}_2=\left[v_2\right] \\ B{\overrightarrow{e}}_3=\left[v_1{v}_2{v}_3\right]\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\Rightarrow B{\overrightarrow{e}}_3=\left[v_1.0_{}{v}_2.0v_3.1\right]\Rightarrow B{\overrightarrow{e}}_3=\left[v_3\right] \end{gathered}$$

Final answer.

$$\begin{gathered} a.A{\overrightarrow{e}}_1=\left[\left.\begin{array}{c}a\\ d\\ g\end{array}\right],A{\overrightarrow{e}}_2=\left[\begin{array}{c}b\\ e\\ h\end{array}\right],A{\overrightarrow{e}}_3=\left[\begin{array}{c}c\\ f\\ k\end{array}\right.\right] \\ b.B{\overrightarrow{e}}_1=\left[v_1\right],B{\overrightarrow{e}}_2=\left[v_2\right],B{\overrightarrow{e}}_3=\left[v_3\right] \end{gathered}$$