Solved problems on linear algebra and vector spaces

1.
(a) c* (x1, x2) = (0, cx2)
let say, c=2, x1= i+j, x2= 2i-3j
L.H.S. = c*(x1, x2)
= 2*(i+j, 2i-3j)
= (2i+2j, 4i-6j)
R.H.S.= (0, cx2)
= (0, 2*(2i-3j))
= (0, 4i-3j)
Here, L.H.S.≠ R.H.S.
Hence we can say that c* (x1, x2) = (0, cx2) is not always true.
(b) c * (X1, x2) = (cx1, c2x2)
Let’s say, c = 3, x1 = i-2j, x2= 4i-j
L.H.S. = c * (X1, x2)
= 3*(i-2j, 4i-j)
= (3i-6j, 12i-3j)
R.H.S. = (cx1, c2x2)
= (3(i-2j), 9(4i-j))
= (3i-6j, 36i-9j)
Here, L.H.S.≠ R.H.S.
Hence we can say that c * (X1, x2) = (cx1, c2x2) is not always true.
(c) c * (x1, x2) = { ( x 1 , x 2 ) if c=1
( cx 1 , cx 2 ) otherwise
This axiom is always true.
C*(x1, x2) = (cx1, cx2)
At c=1,
C*(x1, x2) = 1*(x1,x2) = (x1, x2)
2.
(a) Let X be a non-empty set and F a field.
Vector pointwise operations are:
Sum: f + g  f(x) + g(x)
Product: α(f)  αf(x)
Therefore, set of all functions from x to F follows the vector rules, hence it is a vector space.