Consider the linear transformation $T:R^3->R^3$ whose matrix in relation to the canonical basis is:
$[T] = \begin{bmatrix}1&2&-1\\0&2&3\\1&-1&1\end{bmatrix}$
What is the image of transformation $T$ of the subspace $x+y+2z=0$ in $R^3$?
I know the answer should be the equation of a plane, but I cannot come with the correct answer.
Options are:
How should this be solved?
On
As I explained recently in this answer, writing the equation of the plane as $\mathbf n^T\mathbf x = 0$, if you have the point transformation $\mathbf x'=M\mathbf x$, then the transformed equation is $(M^{-T}\mathbf n)^T\mathbf x'=0$. So, apply $M^T$ to each of the coefficient vectors of the potential solutions until you find one that is a multiple of $(1,1,2)^T$. Don’t look for strict equality since both sides of these equations can be multiplied by a nonzero scalar without changing the planes that they represent. You have to allow for this possibility.
Pick three points on the plane, $$x+y+2z=0$$ and transform the points with your matrix. Now find the equation of a plane passing the three transformed points.
I picked three points $$(0,0,0), (1,1,-1),(2,0,-1)$$ and the transformed points are $$(0,0,0),(4,-1,-1),(3,-3,1)$$ which gives me the plane $$4x+7y+9z=0$$
Thus the correct option is $4$