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Given the basis $\beta = \{(1,-1,3),(-3,4,9),(2,-2,4)\}$ and $x = (8, -9, 6)$, I am to find the corresponding coordinate vector $[x]_\beta$. I claim that the coordinate vectors entries $x_1,x_2,x_3$ meet the following criterion:

$$x_1(1,-1,3)+x_2(-3,4,9)+x_3(2,-2,4) = (8,-9,6)$$ This is equivalent to solving the augmented matrix

\begin{bmatrix} 1 &-3 &2 & 8\\-1 & 4 & -2 & -9\\3 & 9 & 4 & 6 \end{bmatrix}

which is row equivalent to

\begin{bmatrix} 1 &-3 &2 & 8\\0 & 1 & 0 & -1\\0 & 0 & -1 & 0 \end{bmatrix}

which gives $x_3 = 0$, $x_2 = -1$ and $x_1 = 3x_2 + 8 = 5$, thus the coordinate vector $(5, -1, 0)$

There is an error here, seeing as the text claims a different answer. What is the error? In particular, is it computational or simply an error in my understanding of the question?

Edit: fix typo in equation

Community
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Andrew Thompson
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1 Answers1

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Since $$ 5\cdot \pmatrix{1\\-1\\3}+(-1)\cdot\pmatrix{-3\\4\\9} = \pmatrix{5\\-5\\15}+\pmatrix{3\\-4\\-9} = \pmatrix{8\\-9\\6}, $$ your solution is correct.

Maybe you copied the wrong numbers from your textbook?

Christoph
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    Embarassing, but yes, I did. As a more experienced user: should I delete this question or should I let it stay as a "finding coordinate vectors done right" type of thing? – Andrew Thompson Mar 03 '14 at 12:49
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    It's a perfectly valid question and, most of all, shows your own effort! – Christoph Mar 03 '14 at 12:54
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    @AndrewThompson Thanks for keeping this up :) It was actually helpful to me when learning about coordinate vectors with respect to bases - especially because you didn't make any errors! – Burt Aug 23 '20 at 19:42