Suppose for and is a linear transformation.
Domain
Codomain
Find a formula for . Your answer should be a coordinate vector with the variables and in its components.
Find the matrix for the linear transformation (relative to the standard basis in the domain and codomain). That is, find the matrix such that . For instance, enter [ [1,2], [3,4] ] for the matrix .
Find the kernel of . Enter your answer as a vector with constant entries, a vector with the variables or (or both) in its components (using a minimum number of variables), or enter R^2 for .
Find the image of . Enter your answer as a vector with constant entries, a vector with the variables or (or both) in its components (using a minimum number of variables), or enter R^2 for .
The linear transformation is (select all that apply):
A.
bijective (an isomorphism)
B.
surjective (onto)
C.
injective (one-to-one)
D.
none of these
You can earn partial credit on this problem.