Vector equations

Given vectors

$\displaystyle \mathbf{a}_{1} =
\left[\begin{array}{cccc}
a_{11} \\
a_{21} \\
...
...in{array}{cccc}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right],$    and $\displaystyle \mathbf{b} =
\left[\begin{array}{cccc}
b_1 \\
b_2 \\
\vdots \\
b_m \\
\end{array}\right],
$

we can construct a vector equation

$\displaystyle x_1 \mathbf{a}_{1} + \cdots + x_n \mathbf{a}_{n} = \mathbf{b} .$ (1)

It can be solved by using the augmented matrix

$\displaystyle \left[\begin{array}{cccc}
a_{11} & \cdots\cdots & a_{1n} & b_1 \\...
... & \vdots & \vdots \\
a_{m1} & \cdots\cdots & a_{mn} & b_m
\end{array}\right]
$

If the vector equation has a solution (or general solutions), $ \mathbf{b}$ is said to be a linear combination of $ \mathbf{a}_{1}, \ldots, \mathbf{a}_{n}$.

EXAMPLES 2. Let $ \mathbf{a}_1 =
\left[\begin{array}{c}
1 \\
-2 \\
-5
\end{array}\right]$, $ \mathbf{a}_2 =
\left[\begin{array}{c}
2 \\
5 \\
6
\end{array}\right]$, and $ \mathbf{b} =
\left[\begin{array}{c}
7 \\
4 \\
-3
\end{array}\right]$. Determine whether $ \mathbf{b}$ can be generated (or written) as a linear combination of $ \mathbf{a}_1$ and $ \mathbf{a}_2$.



Department of Mathematics
Last modified: 2005-10-21