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

${\begin{aligned}A\mathbf {v} _{1}&=1\mathbf {v} _{1}\\A\mathbf {v} _{2}&=2\mathbf {v} _{2}\end{aligned}}$

Therefore

$T(\mathbf {e} _{1})=A\mathbf {e} _{1}=A\mathbf {v} _{1}-A\mathbf {v} _{2}=\mathbf {v} _{1}-2\mathbf {v} _{2}={\begin{bmatrix}1\\-1\\-1\end{bmatrix}}.$