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Admiral Ackbar's question at Yahoo! Answers (Inverse image of a vector)

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Here is the question:

Could someone explain this? I need to know it for a test, so it would be great if anyone could help.

A linear transformation T: ℝ^3 --> ℝ^3 has matrix
A =
[ 1 -3 1 ]
[ 2 -8 8 ]
[-6 3 -15 ]
Find a vector v in ℝ^3 that satisfies T(v) = [4 -2 9]^T .
Here is a link to the question:

Find vector that satisfies the linear transformation, linear algebra question, PLEASE HELP? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello Admiral Ackbar,

The determinant of the given matrix $A$ is $\det A=108$, so is invertible. We have $$Av=\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}\Leftrightarrow v=A^{-1}\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}=\begin{bmatrix}{1}&{-3}&{1}\\{2}&{-8}&{8}\\{-6}&{3}&{-15}\end{bmatrix}^{-1}\begin{bmatrix}{4}\\{-2}\\{9}\end{bmatrix}=\ldots$$