Question

Consider the basis$\mathfrak{I}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{ℝ}}^{\mathbf{2}}$consisting of the vectors$\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{}\mathbf{and}\mathbf{}\left[\begin{array}{c}3\\ 4\end{array}\right]$

.We are told that${\left[\stackrel{\rightharpoonup }{x}\right]}_{\mathbf{I}}\mathbf{=}\left[\begin{array}{c}7\\ 11\end{array}\right]$for a certain vector$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{ℝ}}^{\mathbf{2}}$

Find

Short answer

If, $\mathfrak{I}of{\mathrm{ℝ}}^2$is a basis consisting of the vectors $\left[\begin{array}{c}1\\ 2\end{array}\right]\mathrm{and}\left[\begin{array}{c}3\\ 4\end{array}\right]\mathrm{and}{\left[\stackrel{\rightharpoonup }{\mathrm{x}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}7\\ 11\end{array}\right]$for a certain vector$\stackrel{\rightharpoonup }{\mathrm{x}}\mathrm{in}{\mathrm{ℝ}}^2$

Then,$\stackrel{\rightharpoonup }{x}=\left[\begin{array}{c}40\\ 58\end{array}\right]$

Step-by-step solution

  Mentioning the concept

Consider a basis $\mathfrak{I}=\left\{{\stackrel{\rightharpoonup }{v}}_1,{\stackrel{\rightharpoonup }{v}}_2,{\stackrel{\rightharpoonup }{v}}_3,...,{\stackrel{\rightharpoonup }{v}}_n\right\}$of a subspace V of ${\mathrm{ℝ}}^n$Then any vector $x\in {\mathrm{ℝ}}^n$can be written uniquely as-

role="math" localid="1660714253890" $$\begin{gathered} \stackrel{\rightharpoonup }{x}=c_1{\stackrel{\rightharpoonup }{v}}_1+c_2{\stackrel{\rightharpoonup }{v}}_2+...c_n{\stackrel{\rightharpoonup }{v}}_n \\ \mathrm{where}\mathrm{the}\mathrm{scalars}{\mathrm{c}}_1,{\mathrm{c}}_2,...,{\mathrm{c}}_{\mathrm{n}}\mathrm{are}\mathrm{called}\mathrm{the}\mathfrak{I}\mathrm{coordinates}\mathrm{of}\mathrm{vector}\stackrel{\rightharpoonup }{\mathrm{x}} \end{gathered}$$

And the vector $\left[\begin{array}{c}{c}_1\\ c_2\\ .\\ .\\ c_n\end{array}\right]$is the $\mathfrak{I}$-coordinate vector of the vector $\stackrel{\rightharpoonup }{x}$denoted by ${\left[\stackrel{\rightharpoonup }{x}\right]}_{\mathfrak{I}}$

$$\begin{gathered} \mathrm{Also},\stackrel{\rightharpoonup }{\mathrm{x}}=\mathrm{S}{\left[\stackrel{\rightharpoonup }{\mathrm{x}}\right]}_{\mathfrak{I}} \\ \mathrm{Where}\mathrm{S}=\left[{\stackrel{\rightharpoonup }{\mathrm{v}}}_1{\stackrel{\rightharpoonup }{\mathrm{v}}}_2{\stackrel{\rightharpoonup }{\mathrm{v}}}_3,....,{\stackrel{\rightharpoonup }{\mathrm{v}}}_{\mathrm{n}}\right] \end{gathered}$$

 Finding the vector x⇀

$$\begin{gathered} \mathrm{We}\mathrm{have}\mathrm{given}\mathrm{a}\mathrm{basis}\mathfrak{I}=\left\{{\stackrel{\rightharpoonup }{\mathrm{v}}}_1,{\stackrel{\rightharpoonup }{\mathrm{v}}}_2\right\}\mathrm{where}{\stackrel{\rightharpoonup }{\mathrm{v}}}_1=\left[\begin{array}{c}1\\ 2\end{array}\right]\mathrm{and}{\stackrel{\rightharpoonup }{\mathrm{v}}}_2=\left[\begin{array}{c}3\\ 4\end{array}\right]. \\ \mathrm{Then}\stackrel{\rightharpoonup }{\mathrm{x}}=\mathrm{S}{\left[\stackrel{\rightharpoonup }{\mathrm{x}}\right]}_{\mathfrak{I}} \\ \because {\left[\stackrel{\rightharpoonup }{\mathrm{x}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}7\\ 11\end{array}\right]\mathrm{and}\mathrm{S}=\left[{\stackrel{\rightharpoonup }{\mathrm{v}}}_1{\stackrel{\rightharpoonup }{\mathrm{v}}}_2\right] \\ \Rightarrow \mathrm{S}=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right] \\ \Rightarrow \stackrel{\rightharpoonup }{\mathrm{x}}=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right] \\ \Rightarrow \stackrel{\rightharpoonup }{\mathrm{x}}=\left[\begin{array}{c}7+33\\ 14+44\end{array}\right] \\ \Rightarrow \stackrel{\rightharpoonup }{\mathrm{x}}=\left[\begin{array}{c}40\\ 58\end{array}\right] \end{gathered}$$

 Final Answer

$$\begin{gathered} \mathrm{If},\mathfrak{I}of{\mathrm{ℝ}}^2\mathrm{is}\mathrm{a}\mathrm{basis}\mathrm{consisting}\mathrm{of}\mathrm{the}\mathrm{vectors}\left[\begin{array}{c}1\\ 2\end{array}\right]\mathrm{and}\left[\begin{array}{c}3\\ 4\end{array}\right]\mathrm{and}{\left[\stackrel{\rightharpoonup }{\mathrm{x}}\right]}_{\mathfrak{I}}=\left[\begin{array}{c}7\\ 11\end{array}\right]\mathrm{for}\mathrm{a} \\ \mathrm{certain}\mathrm{vector}\stackrel{\rightharpoonup }{\mathrm{x}}\mathrm{in}{\mathrm{ℝ}}^2 \\ \mathrm{Then},\stackrel{\rightharpoonup }{\mathrm{x}}=\left[\begin{array}{c}40\\ 58\end{array}\right] \end{gathered}$$