- #1

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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] = b

_{1}[tex]\beta_{1}[/tex] + b

_{2}[tex]\beta_{2}[/tex] + b

_{3}[tex]\beta_{3}[/tex] = d

_{1}[tex]\delta_{1}[/tex] + d

_{2}[tex]\delta_{2}[/tex] + d

_{3}[tex]\delta_{3}[/tex]. Where do I go from there?