I'm stuck on how to begin this problem...
Let (c1, c2) be a solution to the 2x2 system
A11X1 + A12X2 = 0
A21X1 + A22X2 = 0
Show that for any real number alpha (a) the ordered pair (ac1, ac2) is also a solution.
I'm stuck on how to begin this problem...
Let (c1, c2) be a solution to the 2x2 system
A11X1 + A12X2 = 0
A21X1 + A22X2 = 0
Show that for any real number alpha (a) the ordered pair (ac1, ac2) is also a solution.
Plug in $\displaystyle (ac_1,ac_2)$...
$\displaystyle
A_{11}(ac_1)+A_{12}(ac_2)$
$\displaystyle =a\left[A_{11}(c_1)+A_{12}(c_2)\right]$
$\displaystyle =a[0]$
$\displaystyle =0$
Therefore, if $\displaystyle A_{11}(c_1)+A_{12}(c_2)=0$, then $\displaystyle (ac_1,ac_2)$ is also a solution to $\displaystyle A_{11}x_1+A_{12}x_2=0$.
then repeat the process of the second equation. This process can be done more quickly if you use matrix notation:
$\displaystyle \vec{c}$ is a solution to $\displaystyle A\vec{x}=\vec{0}$
$\displaystyle \rightarrow A\vec{c}=\vec{0}$
so, if we test $\displaystyle a\vec{c}$...
$\displaystyle A(a\vec{c})=aA\vec{c}=a\vec{0}=\vec{0}$
therefore $\displaystyle a\vec{c}$ is also a solution to $\displaystyle A\vec{x}=\vec{0}$