Question

Consider the basis$\mathfrak{J}$of ${\mathbf{ℝ}}^{\mathbf{2}}$consisting of the vectors$\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{1}\end{array}\mathbf{\right]}$and $\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\end{array}\mathbf{\right]}\mathbf{,}$let$\mathfrak{R}$be the basis consisting of$\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\end{array}\mathbf{\right]}$$\mathbf{and}\mathbf{\left[}\begin{array}{c}\mathbf{3}\\ \mathbf{4}\end{array}\mathbf{\right]}$Find a matrix P such that

role="math" localid="1660636544224" ${\mathbf{\left[}\overrightarrow{\mathbf{x}}\mathbf{\right]}}_{\mathbf{R}}\mathbf{=}\mathit{P}{\mathbf{\left[}\overrightarrow{\mathbf{x}}\mathbf{\right]}}_{\mathbf{J}}$

for all $\overrightarrow{\mathbf{x}}$in ${\mathbf{ℝ}}^{\mathbf{2}}$.

Short answer

If the basis $\mathfrak{I}\mathrm{of}{\mathrm{R}}^{\mathrm{n}}$consists of the vectors $\left[\begin{array}{c}1\\ 1\end{array}\right]\mathrm{and}\left[\begin{array}{c}1\\ 2\end{array}\right]$be the basis consisting of $\left[\begin{array}{c}1\\ 2\end{array}\right]\mathrm{and}\left[\begin{array}{c}3\\ 4\end{array}\right]$if there exists a matrix P such that

localid="1660715590791" $$\begin{gathered} {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=P\left[\overrightarrow{x}\right]\mathfrak{I}. \\ \Rightarrow P=\frac{1}{2}\left[\begin{array}{cc}-1& 2\\ 1& 0\end{array}\right] \end{gathered}$$

Step-by-step solution

  Mentioning the concept

Consider a basis$\mathfrak{I}=\left\{{\overrightarrow{v}}_{1,}{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,.....,{\overrightarrow{v}}_n\right\}$ of a subspace V. Then any vector $x\in R^n$can be written uniquely as-

role="math" localid="1660714328508" $$\begin{gathered} \overrightarrow{\mathrm{x}}={\mathrm{c}}_1{\overrightarrow{\mathrm{v}}}_1+{\mathrm{c}}_2{\overrightarrow{\mathrm{v}}}_2+...+{\mathrm{c}}_{\mathrm{n}}{\overrightarrow{\mathrm{v}}}_{\mathrm{n}} \\ \mathrm{where}\mathrm{the}\mathrm{scalars}{\mathrm{c}}_1,{\mathrm{c}}_2,....,{\mathrm{c}}_{\mathrm{n}}\mathrm{are}\mathrm{called}\mathrm{the}\mathfrak{I}\mathrm{coordinates}\mathrm{of}\mathrm{vectors}\overrightarrow{\mathrm{x}} \end{gathered}$$

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 $\overrightarrow{x}$denoted by${\left|\overrightarrow{x}\right|}_{\mathfrak{I}}$

Also,$\overrightarrow{x}=S{\left[\overrightarrow{x}\right]}_{\mathfrak{I}}$

Where$S=\left[{\overrightarrow{v}}_1{\overrightarrow{v}}_2{\overrightarrow{v}}_3,.....,{\overrightarrow{v}}_n\right]$

 Finding 

Let’s take the bases$\mathfrak{I}=\left\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2\right\}\mathrm{and}\mathfrak{R}=\left\{{\overrightarrow{w}}_1,{\overrightarrow{w}}_2\right\}$ where

${\overrightarrow{v}}_1=\left[\begin{array}{c}1\\ 1\end{array}\right],{\overrightarrow{v}}_2=\left[\begin{array}{c}1\\ 2\end{array}\right],{\overrightarrow{w}}_1=\left[\begin{array}{c}1\\ 2\end{array}\right],{\overrightarrow{w}}_2=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Let $S_1=\left[{\overrightarrow{v}}_1,{\overrightarrow{v}}_2\right]\mathrm{and}{S}_2=\left[{\overrightarrow{w}}_1,{\overrightarrow{w}}_2\right]$

$S_1=\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]\mathrm{and}{S}_2=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

$S_2^{-1}=\frac{-1}{2}\left[\begin{array}{cc}4& -3\\ -2& 1\end{array}\right]$

Now we have,role="math" localid="1660715187593" $\overrightarrow{x}=S_1{\left[\overrightarrow{x}\right]}_{\mathfrak{I}}\mathrm{and}\overrightarrow{x}=S_2{\left[\overrightarrow{x}\right]}_{\mathfrak{R}}$

$$\begin{gathered} \Rightarrow S_2{\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=S_1{\left[\overrightarrow{x}\right]}_{\mathfrak{I}} \\ \Rightarrow {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=S_2^{-1}{S}_1{\left[\overrightarrow{x}\right]}_{\mathfrak{I}} \end{gathered}$$

Finding the P matrix

Since,$\overrightarrow{x}=S_1{\left[\overrightarrow{x}\right]}_{\mathfrak{I}}\mathrm{and}\overrightarrow{x}=S_2{\left[\overrightarrow{x}\right]}_{\mathfrak{R}}$

$$\begin{gathered} \Rightarrow {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=\frac{-1}{2}\left[\begin{array}{cc}4& -3\\ -2& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]{\left[\overrightarrow{x}\right]}_{\mathfrak{I}} \\ \Rightarrow {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=\frac{-1}{2}\left[\begin{array}{cc}1& -2\\ -1& 0\end{array}\right]{\left[\overrightarrow{x}\right]}_{\mathfrak{I}} \\ \Rightarrow {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=\frac{1}{2}\left[\begin{array}{cc}-1& 2\\ 1& 0\end{array}\right]{\left[\overrightarrow{x}\right]}_{\mathfrak{I}} \\ \Rightarrow P=\frac{1}{2}\left[\begin{array}{cc}-1& 2\\ 1& 0\end{array}\right] \end{gathered}$$

Final Answer

If the basis $\mathfrak{I}\mathrm{of}{R}^n$consists of the vectors $\left[\begin{array}{c}1\\ 1\end{array}\right]\mathrm{and}\left[\begin{array}{c}1\\ 2\end{array}\right]$be the basis consisting of role="math" localid="1660715569674" $\left[\begin{array}{c}1\\ 2\end{array}\right]\mathrm{and}\left[\begin{array}{c}3\\ 4\end{array}\right]$if there exists a matrix P such that

$$\begin{gathered} {\left[\overrightarrow{x}\right]}_{\mathfrak{R}}=P\left[\overrightarrow{x}\right]\mathfrak{I}. \\ \Rightarrow P=\frac{1}{2}\left[\begin{array}{cc}-1& 2\\ 1& 0\end{array}\right] \end{gathered}$$