How can I find a linear transformation matrix (standard matrix) of T: $x = ±1 \rightarrow x^2-y^2=1$. If there is no such linear transformation matrix, I need to explain why.
2026-03-25 09:37:14.1774431434
Linear Transformation of line to hyperbola
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No, there is no such linear transformation. Since we are working in a plane, we have to consider $2\times 2$ matrices. Now, if our matrix has not full rank, then it's image is a vector subspace of dimension $1$ or $0$, but a hyperbola is not a subset of any such vector subspace.
If our matrix has full rank (i.e. it's invertible) then it induces a particular affine transformation(an affine transformation with translation vector $ =$ zero vector), and all affine transformation send every affine subspace in other affine subspace of the same dimension.
Now, a hyperbola is not the union of two straigh lines, so there is no matrix which send those two lines into a hyperbola.