Final answer:
The transformation associated with the matrix \( = \{[1,1], [0,-1]\} \) is \( T(x, y) = (x + y, -y) \). This represents adding the y-coordinate to the x-coordinate and negating the y-coordinate.
Step-by-step explanation:
To find a transformation associated with the matrix \( = \{[1,1], [0,-1]\}, we need to determine how this matrix acts on a given vector in two-dimensional space. This matrix represents a linear transformation of the coordinate plane.
Any vector \( \textbf{v} = (x, y) \) in the plane, when multiplied by this matrix, will be transformed to a new vector \( \textbf{v}' \) calculated as follows:
- For the x-component: \( 1 \cdot x + 1 \cdot y \)
- For the y-component: \( 0 \cdot x + (-1) \cdot y = -y \)
So the transformation \( T\) that this matrix represents can be written as:
\( T(x, y) = (x + y, -y) \)
This transformation consists of adding the y-coordinate to the x-coordinate and negating the y-coordinate.