Section II.1.1 Written Homework Problems
MTH 214 - Linear Algebra
1. Suppose V = {x ∈ R : x > 0}. Define the following operations on V : If x, y ∈ V and r ∈ R, then
• Addition. x ⊕ x = xy (regular multiplication)
• Multiplication. r x = xr (regular exponentiation)
We use the symbols ⊕ and so we don’t confuse them with the symbols of regular addition and multiplication.
Show that V with the operations of ⊕ and forms a vector space by verifying the ten vector space axioms.
2. Suppose V = {S}, a set with only one element. Define the following operations:
• Addition. S + S = S
• Multiplication. rS = S for a real scalar r
Is V a vector space with these operations? Justify your answer.
3. Suppose V = {f : R → R : f (0) = 1} with the operations
• Addition. (f + g)(x) = f (x) + g(x)
• Multiplication. (r · f )(x) = r f (x)
Is V a vector space with these operations? Justify your answer.
Spring 2017
MTH 214 - Prof. Lew
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