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Change of Basis

  • Thread starter QuantumP7
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Homework Statement

We are given 2 bases for V = [tex]\Re_{1 x 3}[/tex]. They are
[tex]\beta_{1}[/tex] = [tex] \begin{bmatrix} 2 & 3 & 2\end{bmatrix} [/tex]

[tex]\beta_{2}[/tex] = [tex] \begin{bmatrix} 7 & 10 & 6\end{bmatrix} [/tex]

[tex]\beta_{3}[/tex] = [tex] \begin{bmatrix} 6 & 10 & 7\end{bmatrix} [/tex]

and,

[tex]\delta_{1}[/tex] = [tex] \begin{bmatrix} 1 & 1 & 1\end{bmatrix} [/tex]

[tex]\delta_{2}[/tex] = [tex] \begin{bmatrix} 0 & 1 & 1\end{bmatrix} [/tex]

[tex]\delta_{3}[/tex] = [tex] \begin{bmatrix} 1 & 1 & 0\end{bmatrix} [/tex]

we are asked to find the [tex]\beta[/tex] to [tex]\delta[/tex] change of basis matrix.

The book says "by solving the relevant system of equations," you get

[tex]\beta_{1}[/tex] = [tex]\delta_{1}[/tex] + [tex]\delta_{2}[/tex] + [tex]\delta_{3}[/tex]

[tex]\beta_{2}[/tex] = 3[tex]\delta_{1}[/tex] + 3[tex]\delta_{2}[/tex] + 4[tex]\delta_{3}[/tex]

[tex]\beta_{3}[/tex] = 3[tex]\delta_{1}[/tex] + 4[tex]\delta_{2}[/tex] + 3[tex]\delta_{3}[/tex]


My question is WHAT system of equations did they solve to get the above?! I'm at a complete loss.

Homework Equations





The Attempt at a Solution


I know that for any vector [tex]\alpha[/tex], [tex]\alpha[/tex] = b1[tex]\beta_{1}[/tex] + b2[tex]\beta_{2}[/tex] + b3[tex]\beta_{3}[/tex] = d1[tex]\delta_{1}[/tex] + d2[tex]\delta_{2}[/tex] + d3[tex]\delta_{3}[/tex]. Where do I go from there?
 

Answers and Replies

  • #2
vela
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They solved the 3 vector equations

[tex]\begin{align*}
\beta_1 &= d_{11} \delta_1 + d_{12} \delta_2 + d_{13} \delta 3 \\
\beta_2 &= d_{21} \delta_1 + d_{22} \delta_2 + d_{23} \delta 3 \\
\beta_3 &= d_{31} \delta_1 + d_{32} \delta_2 + d_{33} \delta 3
\end{align*}
[/tex]
 

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