I have this problem in linear algebra in the linear transformation section. where T(v)=Av
A=
l 1 3 1 l
l 2 0 0 l
(A=2x3 matrix, first row 1,3,1;second row 2,0,0)
I need help finding the preimage of w=(6,4)
thanks
I have this problem in linear algebra in the linear transformation section. where T(v)=Av
A=
l 1 3 1 l
l 2 0 0 l
(A=2x3 matrix, first row 1,3,1;second row 2,0,0)
I need help finding the preimage of w=(6,4)
thanks
If I understand you right, you need the solution to
$\displaystyle \left ( \begin{array}{c} 6 \\ 4 \end{array} \right ) = \left ( \begin{array}{ccc} 1 & 3 & 1 \\ 2 & 0 & 0 \end{array} \right ) \cdot \left ( \begin{array}{c} x \\ y \\ z \end{array} \right )$
So we know that:
$\displaystyle 6 = x + 3y + z$
$\displaystyle 4 = 2x$
From the second equation we see that x = 2.
Thus the first equation reads:
$\displaystyle 4 = 3y + z$
or
$\displaystyle z = 4 - 3y$
So your preimage is of the form:
$\displaystyle \left ( \begin{array}{c} 2 \\ y \\ 4 - 3y \end{array} \right )$
-Dan