Linear Algebra Done Right · Problem 1B.6

Finite-Dimensional Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · vector-space-definition

Problem

Let ∞\infty and −∞-\infty denote two distinct objects, neither of which is in R\mathbf{R}. Define an addition and scalar multiplication on R∪{∞,−∞}\mathbf{R} \cup \{\infty, -\infty\} as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for t∈Rt \in \mathbf{R} define t∞={−∞if t<0,0if t=0,∞if t>0,t(−∞)={∞if t<0,0if t=0,−∞if t>0,t\infty = \begin{cases} -\infty & \text{if } t < 0, \\ 0 & \text{if } t = 0, \\ \infty & \text{if } t > 0, \end{cases} \qquad t(-\infty) = \begin{cases} \infty & \text{if } t < 0, \\ 0 & \text{if } t = 0, \\ -\infty & \text{if } t > 0, \end{cases} and t+∞=∞+t=∞+∞=∞,t+(−∞)=(−∞)+t=(−∞)+(−∞)=−∞,∞+(−∞)=(−∞)+∞=0.\begin{aligned} t + \infty &= \infty + t = \infty + \infty = \infty,\\ t + (-\infty) &= (-\infty) + t = (-\infty) + (-\infty) = -\infty,\\ \infty + (-\infty) &= (-\infty) + \infty = 0. \end{aligned} With these operations of addition and scalar multiplication, is R∪{∞,−∞}\mathbf{R} \cup \{\infty, -\infty\} a vector space over R\mathbf{R}? Explain.

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