# Writing a vector as a linear combination.

• Sep 23rd 2009, 11:27 AM
GreenDay14
Writing a vector as a linear combination.
The question is:

Write (1, 2, 3) belongs to R3 as a linear combination of (1, 1, 1), (1, 0, 1), (1, 0, 0).

So i wrote:

(1, 2, 3) = 2(1, 1, 1) + 1(1, 0, 1) - 2(1, 0, 1)

Which would give:

2 2 2
+1 0 1
= (3, 2, 3)
- (2, 0, 2)
= (1, 2, 3)

So it would appear that it works out, but it seems way to easy. Could someone check this for me and let me know if i am missing something? Thanks.
• Sep 23rd 2009, 12:22 PM
Plato
Quote:

Originally Posted by GreenDay14
The question is:
Write (1, 2, 3) belongs to R3 as a linear combination of (1, 1, 1), (1, 0, 1), (1, 0, 0).
So i wrote:
(1, 2, 3) = 2(1, 1, 1) + 1(1, 0, 1) - 2(1, 0, 1)

That is the correct solution. It could have been found by solving
$\left( {\begin{array}{*{20}c}
1 & 1 & 1 \\
1 & 0 & 0 \\
1 & 1 & 0 \\
\end{array} } \right)\left( {\begin{array}{*{20}c} x \\ y \\ z \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
1 \\ 2 \\ 3 \\ \end{array} } \right)$