Originally Posted by
jsndacruz I am currently learning about linear transforms and came across this problem:
Given a matrix: A =
[ 4 3 -1 7 ]
[ 1 1 0 2 ]
[ 2 5 3 7 ]
We are told that to find all the values t such that b = [ 1 t t^2 ]^T is solvable.
My attempt: I augmented the matrix to,
[ 4 3 -1 7 | 1 ]
[ 1 1 0 2 | t ]
[ 2 5 3 7 | t^2]
and ended up with:
[ 1 0 0 0 | t^2 -6t +1 ]
[ 0 1 2 0 | t^2 + t - 1 ]
[ 0 0 1 -1| t^2 - 3t ]
I'm not sure how to proceed from here. I understand that x_1 = t^2 - 6t + 1, that x_2 + 2 x_3 = t^2 + t - 1, and that x_3 - x_4 = t^2 - 3t. But how do those equations affect the constraints on t? It seems that no matter what I plug in for my free variable x_4, I can plug in whatever I want for t.