Final answer:
To find the standard matrix for a transformation, apply the transformation equations to the standard basis vectors and record the resulting vectors. These resulting vectors will form the columns of the standard matrix.
Step-by-step explanation:
The student is asking to find the standard matrix for a transformation defined by the equations. To find the standard matrix for a transformation, we need to determine the images of the standard basis vectors under the transformation. The standard basis vectors are a set of vectors with only one non-zero entry, which represents the coordinate axes. We apply the transformation equations to each standard basis vector and record the resulting vector. These resulting vectors will form the columns of the standard matrix.
For example, if the transformation is defined by the equations x' = x cos(q) + y sin(o) and y' = -x sin(p) + y cos(p), we can consider the standard basis vectors Î and Ĵ. When we apply the transformation equations to Î, we get (1)Î = (1)(cos(q)) + (0)(sin(o)) = cos(q). When we apply the transformation equations to Ĵ, we get (1)Ĵ = (0)(cos(p)) + (1)(cos(p)) = sin(p). Therefore, the standard matrix for this transformation is:
[ cos(q) sin(p) ]
[ -sin(p) cos(p) ]