Given the full and correct answer the two bases of B1

Given the full and correct answer the two bases of B1

Question
Alternate coordinate systems
asked 2020-10-18

Given the full and correct answer the two bases of \(B1 = \left\{ \left(\begin{array}{c}2\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 2\end{array}\right) \right\}\)
\(B_2 = \left\{ \left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right) \right\}\)
find the change of basis matrix from \(\displaystyle{B}_{{1}}\to{B}_{{2}}\) and next use this matrix to covert the coordinate vector
\(\overrightarrow{v}_{B_1} = \left(\begin{array}{c}2\\ -1\end{array}\right)\) of v to its coodirnate vector
\(\overrightarrow{v}_{B_2}\)

Answers (1)

2020-10-19

\(B_1 = \left\{ \begin{bmatrix}2 \\ 1 \end{bmatrix}, \begin{bmatrix}3 \\ 2 \end{bmatrix} \right\} and B_2 = \left\{ \begin{bmatrix}3 \\ 1 \end{bmatrix}, \begin{bmatrix}7 \\ 2 \end{bmatrix} \right\}\) Consider \([B_2 : B_1] = \begin{bmatrix}3 & 7 &2 & 3 \\ 1 & 2 & 1 & 2 \end{bmatrix}\)
\(\sim \begin{bmatrix}1 & 2 & 1 & 2 \\ 3 & 7 & 2 & 3 \end{bmatrix} (R_1 \leftrightarrow R_2)\)
\(\sim \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & -1 & -3 \end{bmatrix} (R_1 \rightarrow R_2 - 3 R_1)\)
\(\sim \begin{bmatrix}1 & 0 & 3 & 8 \\ 0 & 1 & -1 & -3 \end{bmatrix} (R_1 \leftrightarrow R_1 - 2 R_2)\)
\(\displaystyle\therefore\) change of basis matrix from
\(\displaystyle{B}_{{1}}\to{B}_{{2}}{i}{s}\)
\(P = \begin{bmatrix} 3 & 8 \\ -1 & -3 \end{bmatrix}\)
\(men [v]_{B_1} = \begin{bmatrix} 2\\-1 \end{bmatrix}\)
\(men [v]_{B_2} = P[v]_{B_1} = \begin{bmatrix}3 & 8 \\-1 & -3 \end{bmatrix} \begin{bmatrix} 2\\-1 \end{bmatrix}\)
\(\therefore [v]_{B_2} = \begin{bmatrix} -2\\ 1\end{bmatrix}\)

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