Matrix Algebra

Vector Equations

Vector Equation. Given vectors

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

we can construct a vector equation

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

with unknown scalar variables $x_1,\ldots,x_n$ .

Augmented Matrix. The vector equation can be formulated by the following 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 above vector equation has a solution (or general solutions), we say that $\mathbf{b}$ is a linear combination of $\mathbf{a}_{1}, \ldots, \mathbf{a}_{n}$ , and write $\mathbf{b} \in$ span $\{\mathbf{a}_{1}, \ldots, \mathbf{a}_{n}\}$ in short.

EXAMPLES 2. Let $\mathbf{a}_1 = \left[\begin{array}{c} 1 \\ 2 \\ 0 \end{array}\right]$ , $\mathbf{a}_2 = \left[\begin{array}{c} 0 \\ 2 \\ 2 \end{array}\right]$ , and $\mathbf{b} = \left[\begin{array}{c} 0 \\ 6 \\ 4 \end{array}\right]$ . Determine whether $\mathbf{b}$ can be generated (or written) as a linear combination of $\mathbf{a}_1$ and $\mathbf{a}_2$ .

Is $\mathbf{b} = \left[\begin{array}{c} 1 \\ 6 \\ 4 \end{array}\right]$ a linear combination of $\mathbf{a}_1$ and $\mathbf{a}_2$ ?

Geometric Description of Spanned Subsets. Let $\mathbf{a}_1$ and $\mathbf{a}_2$ be nonzero vectors in $\mathbb{R}^3$ . If $\mathbf{a}_2$ is not a multiple of $\mathbf{a}_1$ then $\mathrm{span}\{\mathbf{a}_{1}, \mathbf{a}_{2}\}$ is the plane in $\mathbb{R}^3$ that contains $\mathbf{a}_1$ and $\mathbf{a}_2$ . In particular the plane includes $\mathbf{b}\in\mathrm{span}\{\mathbf{a}_{1}, \mathbf{a}_{2}\}$ ; see the black line on the plane in the figure below. On the other hand when $\mathbf{b}\not\in\mathrm{span}\{\mathbf{a}_{1}, \mathbf{a}_{2}\}$ (that is, $\mathbf{b}$ does not belong to $\mathrm{span}\{\mathbf{a}_{1}, \mathbf{a}_{2}\}$ ) the plane does not contain $\mathbf{b}$ ; in the second figure the black line is not on the plane.