Change of Basis Homework: Solving System of Equations

[QuantumP7]

Homework Statement

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

\beta_{2} = \begin{bmatrix} 7 & 10 & 6\end{bmatrix}

\beta_{3} = \begin{bmatrix} 6 & 10 & 7\end{bmatrix}

and,

\delta_{1} = \begin{bmatrix} 1 & 1 & 1\end{bmatrix}

\delta_{2} = \begin{bmatrix} 0 & 1 & 1\end{bmatrix}

\delta_{3} = \begin{bmatrix} 1 & 1 & 0\end{bmatrix}

we are asked to find the \beta to \delta change of basis matrix.

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

\beta_{1} = \delta_{1} + \delta_{2} + \delta_{3}

\beta_{2} = 3\delta_{1} + 3\delta_{2} + 4\delta_{3}

\beta_{3} = 3\delta_{1} + 4\delta_{2} + 3\delta_{3}


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 \alpha, \alpha = b₁\beta_{1} + b₂\beta_{2} + b₃\beta_{3} = d₁\delta_{1} + d₂\delta_{2} + d₃\delta_{3}. Where do I go from there?

 

[vela]

They solved the 3 vector equations

\begin{align*}<br /> \beta_1 &amp;= d_{11} \delta_1 + d_{12} \delta_2 + d_{13} \delta 3 \\<br /> \beta_2 &amp;= d_{21} \delta_1 + d_{22} \delta_2 + d_{23} \delta 3 \\<br /> \beta_3 &amp;= d_{31} \delta_1 + d_{32} \delta_2 + d_{33} \delta 3<br /> \end{align*}<br />