Take the set of all pairs of real numbers with the usual addition,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

Take the set of all pairs of real numbers (x,y)(x,y) with the usual addition, (x1,y1)+(x2,y2)=(x1+x2, y1+y2)(x_1,y_1)+(x_2,y_2) = (x_1+x_2,\, y_1+y_2), but with a new scaling that stretches only the first entry, α⊙(x,y)=(αx, y)\alpha\odot(x,y) = (\alpha x,\, y) for real α\alpha. ((a)) Show that the closure requirement holds for both operations. ((b)) Test each of the eight rules. For every rule that fails, give a counterexample with numbers; for every rule that holds, give the one-line reason. ((c)) Evaluate 0⊙(1,1)0\odot(1,1). In an ordinary vector space, scaling any member by 00 gives the zero member, because 0u=(0+0)u=0u+0u0\mathbf{u} = (0+0)\mathbf{u} = 0\mathbf{u}+0\mathbf{u}. Say which step of that argument breaks here, and decide whether the set with these operations is a vector space.

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