# Change of Basis

## 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.

## 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?

\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*}