4. (15%) Let \( \mathbf{R} \) denote the set of real numbers. Show that \( \mathbf{R} \), together with the usual addition and scalar multiplication of real numbers, is a vector space.
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4. (15%) Let \( \mathbf{R} \) denote the set of real numbers. Show that \( \mathbf{R} \), together with the usual addition and scalar multiplication of real numbers, is a vector space.