Question

Question: Repeat Exercise 35, assuming u and v are eigenvectors of A that correspond to eigenvalues -1 and 3, respectively.

Short answer

The image is given below:

Step-by-step solution

Plot \(T\left( {\rm{u}} \right) =  - {\rm{u}}\)

With the help of the given figure in the exercise, draw \( - u\) because it is given that \(u\) is an eigenvector of matrix A that corresponds to eigenvalue -1.

Plot \(T\left( {\rm{v}} \right) = 3{\rm{v}}\)

Similarly, with the help of the given figure in the exercise, draw \(3v\) because it is given that \(u\) is an eigenvector of matrix A that corresponds to eigenvalue 3.

Plot \(T\left( {\rm{w}} \right)\)

It is given that \(w = u + v\), since \(T\) is linear transformation given by \(T\left( x \right) = Ax\).

So, \(T\left( w \right)\) can be obtained as follows:

\(\begin{array}{c}T\left( w \right) = T\left( {u + v} \right)\\ = A\left( {u + v} \right)\\ = A\left( u \right) + A\left( v \right)\\ = T\left( u \right) + T\left( v \right)\end{array}\)

Now, the plot \(T\left( w \right) = T\left( u \right) + T\left( v \right)\) as given in the image below: