Problem 35E

Let u and v be the vectors shown in the figure, and suppose u and v are eigenvectors of a 2 × 2 matrix A that correspond to eigenvalues 2 and 3, respectively. Let be the linear transformation given by for each x in R2, and let w = u + v. Make a copy of the figure, and on the same coordinate system, carefully plot the vectors

Solution 35E

Plot as 2u, since u is an eigenvector for the eigenvalue 2 and =

Similarly, plot as 3v, because v is an eigenvector for the eigenvalue 3 and = v= 3v

Now, the plot of the vector as shown below

The graph contains the vectors , , and .