Let V be the set of all positive real numbers. Define vector addition as follows

Question

Let V be the set of all positive real numbers. Define vector addition as follows: given any two positive real numbers x and y, their “sum” is written as "x + y" but defined to be the number xy (ordinary multiplication). So, for example, if x = 3 and y = 5 then "x + y" would be "3+ 5" = 3 5 =15. Define scalar multiplication as follows: given any real scalar r and positive real number x, the “scalar multiple” is "rx" = xr . For example, if r = 2 and x = 3 , then "rx" would be defined as "(2)3" = 32 =1 9 . Explain why V is a vector space with these operations. In other words, check that every one of the properties defining a vector space is true (based on properties of ordinary real numbers). In particular, what is the zero vector in V? For any x in V, what is the opposite vector of x?

Explanation / Answer

V is a vector space with above two operations because both "sum" and "scalar multiplication " results in element of vector. The properties of vector space are satisfied.

The zero vector in V is 0 , as x+0 results in 0 and 0x results in x.

Hence , vector 0 is zero vector of x,

The opposite vector of x is x itself as xx results in x-x=0

hence the each vector is oppisite to itself.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.