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2.1.14. Use Exercise 2.1.13 to show that the space of pairs $(f(x), a)$, where $f$ is a continuous scalar function and $a$ is a real number, is a vector space. What is the zero element? Be precise! Write out the laws of vector addition and scalar multiplication.

   2.1.14. Use Exercise 2.1.13 to show that the space of pairs $(f(x), a)$, where $f$ is a continuous scalar function and $a$ is a real number, is a vector space. What is the zero element? Be precise! Write out the laws of vector addition and scalar multiplication.
 
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Applied Linear Algebra (Undergraduate Texts in Mathematics)
Applied Linear Algebra (Undergraduate Texts in Mathematics)
Peter J. Olver,… 2nd Edition
Chapter 2, Problem 14 ↓

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The space in question consists of pairs $(f(x), a)$, where $f(x)$ is a continuous scalar function from some domain (typically $\mathbb{R}$ or a subset of it) to $\mathbb{R}$, and $a$ is a real number. An element of this space is thus a pair consisting of a  Show more…

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2.1.14. Use Exercise 2.1.13 to show that the space of pairs $(f(x), a)$, where $f$ is a continuous scalar function and $a$ is a real number, is a vector space. What is the zero element? Be precise! Write out the laws of vector addition and scalar multiplication.
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