Science:Math Exam Resources/Courses/MATH152/April 2012/Question 03 (b)/Solution 1

The linear transformation acting on the vector ${\mathbf{𝐞}}_1$ will get given by

$T({\mathbf{𝐞}}_1)=A{\mathbf{𝐞}}_1.$

From part (a) we have that

${\mathbf{𝐞}}_1={\mathbf{𝐯}}_1-{\mathbf{𝐯}}_2.$

In the question we are given the eigenvalues for these eigenvectors so we know

$\begin{array}{rl}A{\mathbf{𝐯}}_1& =1{\mathbf{𝐯}}_1\\ A{\mathbf{𝐯}}_2& =2{\mathbf{𝐯}}_2\end{array}$

Therefore

$T({\mathbf{𝐞}}_1)=A{\mathbf{𝐞}}_1=A{\mathbf{𝐯}}_1-A{\mathbf{𝐯}}_2={\mathbf{𝐯}}_1-2{\mathbf{𝐯}}_2=\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right].$