Character of direct sum of linear representations is sum of characters

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Statement

Suppose G is a group, K is a field, and \rho_1:G \to GL(V_1), \rho_2:G \to GL(V_2) are finite-dimensional linear representations of G over K. Denote by \chi_{\rho_1}, \chi_{\rho_2} the characters of the representations \rho_1,\rho_2 respectively. Denote by \rho_1 \oplus \rho_2 the direct sum of linear representations \rho_1,\rho_2 and by \chi_{\rho_1 \oplus \rho_2} its character. Then, we have the following for any g \in G:

\chi_{\rho_1 \oplus \rho_2} = \chi_{\rho_1}(g) + \chi_{\rho_2}(g)

Recall that the character of a linear representation is the function that sends any g to the trace of the corresponding linear map.

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