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Linear form
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=== Linear functionals in R<sup>''n''</sup> === Suppose that vectors in the real coordinate space <math>\R^n</math> are represented as column vectors <math display=block>\mathbf{x} = \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math> For each row vector <math>\mathbf{a} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix}</math> there is a linear functional <math>f_{\mathbf{a}}</math> defined by <math display=block>f_{\mathbf{a}}(\mathbf{x}) = a_1 x_1 + \cdots + a_n x_n,</math> and each linear functional can be expressed in this form. This can be interpreted as either the matrix product or the dot product of the row vector <math>\mathbf{a}</math> and the column vector <math>\mathbf{x}</math>: <math display=block>f_{\mathbf{a}}(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix} \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>
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