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

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- Nov 20th 2006, 07:30 AMwilkie610Need help finding a preimage (linear algebra)
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 - Nov 20th 2006, 10:36 AMtopsquark
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