C25 – pg. 287
Show that the set W = \[\begin{Bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}| & 3x_{1} - 5x_{2} = 12 \end{Bmatrix}\]from Example NSC2Z fails Property AC and Property SC.
fail AC
let x = \[\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\] suppose that y\(\in\) W and y = \[\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix}\].
Therefore, x+y = \[\begin{bmatrix} x_{1} + y_{1} \\ x_{2} + y_{2} \end{bmatrix}\].
According to the equation in set W, we get a new equation:
\(3(x_{1} + y_{1}) - 5(x_{2}+y_{2})\) ……. new equation
\(= 3x_{1} + 3y_{1} - 5x_{2} - 5y_{2}\) ……. expend the equation
\(= 3x_{1} - 5x_{2} + 3y_{1} - 5y_{2}\) ……. swap element position
\(= 12 + 12\) ……. plug in value
\(= 24 \neq 12\)
we see that 24 \(\neq\) 12 provided in set W, so it fails property AC
fail SC
with the similar structure. we let x = \[\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}\] suppose \(\alpha\)x = \[\begin{bmatrix} \alpha*x_{1} \\ \alpha*x_{2} \end{bmatrix}\]substitute \(\alpha\)x with x in the equation, we get:
3\(\alpha\)\(x_{1}\) - 5\(\alpha\)\(x_{2}\) …….plug \(\alpha\)x in equation
= \(\alpha(3x_{1} - 5x_{2})\) …… apply associative law
= \(\alpha\) * 12
as we can see only when \(\alpha\) = 1, the W can be considered as a subspace of \(C^{3}\) which means that the property O holds. However, property SC fails