Applications

This section shows some application examples based on the Qvious performed.

Block

Example: \[\begin{aligned} \phi(x,y) &= \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right) \\[2em] &= \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) \\[2em] &= (x_1, \ldots, x_n) \left(\begin{array}{ccc} \phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\ \vdots & \ddots & \vdots \\ \phi(e_n, e_1) & \cdots & \phi(e_n, e_n) \end{array}\right) \left(\begin{array}{c} y_1 \\ \vdots \\ y_n \end{array}\right) \end{aligned}\]

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Markdown:

$$
\begin{aligned}
  \phi(x,y) &= \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right) \\[2em]
            &= \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j)            \\[2em]
            &= (x_1, \ldots, x_n)
               \left(\begin{array}{ccc}
                 \phi(e_1, e_1)  & \cdots & \phi(e_1, e_n) \\
                 \vdots          & \ddots & \vdots         \\
                 \phi(e_n, e_1)  & \cdots & \phi(e_n, e_n)
               \end{array}\right)
               \left(\begin{array}{c}
                 y_1    \\
                 \vdots \\
                 y_n
               \end{array}\right)
\end{aligned}
$$

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{:.figcaption}

KaTeX does not support the align and align* environments. Instead, aligned should be used, e.g. \begin{aligned} ... \end{aligned}.