LINEAR ALGEBRA need the complete and clear solution thanks D

LINEAR ALGEBRA need the complete and clear solution thanks Determine whether the set equipped with the standard operations of vector addition and scalar multiplication is a vector space. (10 pts). The set ffxy):y 2x-5) with the standard operations in R3.

Solution

We need to determine whether the given set equipped with the standard operations of vector addition and scalar multiplication is a vector space or not.

Set is given to be {(x,y): Y=2x-5}

Answer:

A set is said to be a vector space if it is not empty and it is closed under vector addition and scalar multiplication.

Emptiness:

Our set is given to be (x, y), where x and y being real. Hence for any value of x and y the given set is never empty (even for (0, 0)), hence we showed that the given set is not empty.

Closeness under Vector Addition:

By closed under vector addition we mean, that if two vectors of the given set are added, then the result of addition as well should lie in the given set!

For any real value of x, we will always get a real value of y (We have y=2x-5. For every real x, y is always real). And sum of two real values (x+y) is always real. Hence the given set is closed under addition.

Closeness of scalar multiplication:

If any of the vectors x and y be multiplied with a real number (scalar), the result of multiplication will also be real because product of two real numbers is always real, hence result of scalar multiplication also lies in the given set. Thus, the give set is closed under scalar multiplication.

Since all the three conditions are satisfied, hence the given set is indeed a vector space!!