Science:Math Exam Resources/Courses/MATH221/April 2009/Question 09 (c)/Solution 1

To find the coordinate vector of ${\begin{bmatrix}0\\1\\2\end{bmatrix}}$relative to the basis of $Col(A)$, we place the basis for $Col(A)$ and the coordinate vector in an augmented matrix

$\left.{\begin{matrix}2&1&\\1&2&\\0&3&\end{matrix}}\right|{\begin{matrix}0\\1\\2\end{matrix}}$

R1-R2 R2-R1 R3-R2 R2 divided by 3 R1+R2

$\left.{\begin{matrix}1&0&\\0&1&\\0&0&\end{matrix}}\right|{\begin{matrix}-1/3\\2/3\\0\end{matrix}}$

so we have x1 = -1/3 x2 = 2/3

Thus, the coordinate vecctor of ${\begin{bmatrix}0\\1\\2\end{bmatrix}}$ relative to basis of $Col(A)$ is ${\binom {-1/3}{2/3}}$