Are any of these vector spaces and if not why not.

Question

a. The set {(x, .5x), where x is a real number} with standard operations for addition and scalar multiplication.

b. The set of all 2x2 matrices of the form [ 𝑎 𝑏]

[0 𝑐 ] where a, b, and c are real numbers. The set has standard operations for addition and scalar multiplication.

c. The set of all elements (x,y), where addition is defined (x1, y1) + (x2, y2)= (x1+x2, y1+y2) and scalar multiplication is defined 𝑐(𝑥1 , 𝑦1 ) = (√𝑐𝑥1 ,√𝑐𝑦1 )

Answer 1

For a, the zeros vector is included in the set (simply take x = 0) Also if you pick u = (x,0.5x) and v = (y,0.5y), you can easily see that u+v = v+u. Also if you pick a in R then a(u+v) = au + av. So, {(x, .5x), where x is a real number} is indeed a vector space.

The format of b and c is not very clear.