- #1
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?