Find a Basis for the Set of All Solutions to the Difference

question 27

Multiple Choice

Find a basis for the set of all solutions to the difference equation.
-Let $B = \left\{ \mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } \right\}$ and $C = \left\{ \mathbf { c } _ { 1 } , \mathbf { c } _ { 2 } \right\}$ be bases for $\mathcal { R } ^ { 2 }$ , where
$$\mathbf { b } _ { 1 } = \left[ \begin{array} { l } 2 \\4\end{array} \right] , \mathbf { b } _ { 2 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 1 } = \left[ \begin{array} { l } 1 \\3\end{array} \right] , \mathbf { c } _ { 2 } = \left[ \begin{array} { r } - 4 \\- 10\end{array} \right]$$
Find the change-of-coordinates matrix from $B$ to $C$ .

• A)
• \(\left[ \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right]\)
• B)
• \(\left[ \begin{array} { l l } 0 & - 1 \\ 1 & - 2 \end{array} \right]\)
• C)
• \(\left[ \begin{array} { r r } 1 & - 4 \\ 3 & - 10 \end{array} \right]\)
• D)
• \(\left[ \begin{array} { l l } - 2 & 1 \\ - 1 & 0 \end{array} \right]\)

---

Definitions:

Equilibrium Price

The price at which the quantity of a good or service demanded by consumers matches the quantity supplied by producers, leading to a balance in the market.

Allocative Efficiency

A state of the economy where resources are distributed in a way that maximizes the net benefit to society, with goods and services going to those who value them the most.

Marginal Cost

The incremental cost incurred when producing one more unit of a good or service. This is a different phrasing for the concept given previously.

Marginal Benefit

The additional satisfaction or utility gained by consuming an extra unit of a good or service, a reiteration with different phrasing.