If the Set W Is a Vector Space, Find a Set

question 14

Multiple Choice

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.
- $W$ is the set of all vectors of the form $\left[ \begin{array} { c } a + 6 b \\ 3 b \\ 4 a - b \\ - a \end{array} \right]$ , where $a$ and $b$ are arbitrary real numbers.

• A)
• \(\left[ \begin{array} { r } 1 \\ 3 \\ 4 \\ - 1 \end{array} \right] , \left[ \begin{array} { r } 6 \\ 0 \\ - 1 \\ 0 \end{array} \right]\)
• B)
• \(\left[ \begin{array} { r } 1 \\ 0 \\ 4 \\ - 1 \end{array} \right] , \left[ \begin{array} { r } 6 \\ 3 \\ - 1 \\ 0 \end{array} \right]\)
• C) Not a vector space
• D)
• \[\left[ \begin{array} { l }
• 1 \\
• 0 \\
• 4 \\
• 0
• \end{array} \right] , \left[ \begin{array} { r }
• 6 \\
• 0 \\
• - 1 \\
• 0
• \end{array} \right] , \left[ \begin{array} { r }
• 0 \\
• 3 \\
• 0 \\
• - 1
• \end{array} \right]\]

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Definitions:

Utility Maximization

The economic principle that individuals seek to obtain the greatest possible satisfaction from their consumption choices, given their income and prices.

Marginal Utility Per Dollar

A measure of the additional satisfaction gained from spending one more dollar on a good or service.

Consumer Equilibrium

The state where the allocation of goods and services aligns with consumer preferences, and the marginal utility per dollar spent is equalized across all goods.

Giffin Good

A type of good for which demand increases as the price increases, contrary to the basic laws of demand in economics.