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Mathematics LibreTexts

6: Linear Maps

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    270

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    As discussed in Chapter 1, one of the main goals of Linear Algebra is the characterization of solutions to a system of \(m \) linear equations in \(n \) unknowns \( x_1, \ldots, x_n \),

    \[ \begin{equation*}
    \left.
    \begin{array}{rl}
    a_{11} x_1 + \cdots + a_{1n} x_n &= b_1\\
    \vdots \qquad \vdots \qquad & \quad \vdots\\
    a_{m1} x_1 + \cdots + a_{mn} x_n &= b_m
    \end{array}
    \right\},
    \end{equation*} \]

    where each of the coefficients \(a_{ij} \) and \(b_i \) is in \(\mathbb{F} \). Linear maps and their properties give us insight into the characteristics of solutions to linear systems.