Question

In Exercises$\mathbf{1}$through 18 , determine whether the vector$\overrightarrow{\mathbf{x}}$is in the span $\mathit{V}$ of the vectors ${\overrightarrow{\mathit{v}}}_{\mathit{1}}\mathit{,}{\overrightarrow{\mathit{v}}}_{\mathit{2}}\mathit{,}\mathit{\cdots }\mathit{,}{\overrightarrow{\mathit{v}}}_{\mathit{m}}$ (proceed “by inspection” if possible, and use the reduced row-echelon form if necessary). If$\overrightarrow{\mathit{x}}$is in $\mathit{V}$ , find the coordinates of $\overrightarrow{\mathit{x}}$ with respect to the basis$\mathfrak{I}\mathbf{=}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{\dots }\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}$of $\mathit{V}$ , and write the coordinate vector $\left[\overrightarrow{x}\right]\mathfrak{I}$.

localid="1664356781673" $\overrightarrow{\mathbf{x}}\mathbf{=}\left[\begin{array}{c}2\\ 3\end{array}\right]\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}\left[\begin{array}{c}1\\ 0\end{array}\right]\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{=}\left[\begin{array}{c}0\\ 1\end{array}\right]$

Short answer

The coordinates of the vector is, ${\overrightarrow{x}}_{\mathfrak{I}}=\left[\begin{array}{c}2\\ 3\end{array}\right]$.

Step-by-step solution

Consider the vectors

The vectors are,

$\overrightarrow{x}=\left[\begin{array}{c}2\\ 3\end{array}\right];{\overrightarrow{v}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right];{\overrightarrow{v}}_2\left[\begin{array}{c}0\\ 1\end{array}\right]$

Check for the coordinates of the vector

The formula is, $\overrightarrow{x}=c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2$.

$$\begin{gathered} \left[\begin{array}{c}2\\ 3\end{array}\right]=c_1\left[\begin{array}{c}1\\ 0\end{array}\right]+c_2\left[\begin{array}{c}0\\ 1\end{array}\right] \\ {c}_1+0c_2,0c_1+c_2=3 \\ {c}_1=2,c_2=3 \end{gathered}$$

Substitute these values in the formula.

$$\begin{gathered} \overrightarrow{x}=c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2 \\ \left[\begin{array}{c}2\\ 3\end{array}\right]=2\left[\begin{array}{c}1\\ 0\end{array}\right]+3\left[\begin{array}{c}0\\ 1\end{array}\right] \\ {\overrightarrow{x}}_{\mathfrak{I}}=\left[\begin{array}{c}2\\ 3\end{array}\right] \end{gathered}$$

Final answer

The coordinates of the vector is,${\overrightarrow{x}}_{\mathfrak{I}}=\left[\begin{array}{c}2\\ 3\end{array}\right]$.