4. Let V be a vector space over a field K. We claim that for every element v of V, we have
0v = O where 0 is the number 0 in K and O is the zero element of V. A proof of this is given
on page 4 of the textbook. By a "proof," we mean a logical deduction of the claim from our
assumptions (in this case, VS1 to VS 8) or from other results we have already proved.
(a) The first line of the proof reads:
$0v + v = 0v + 1v = (0 + 1)v = 1v = v$.
(i) Which of VS 1 to VS 8 is used to justify the first equality: $0v + v = 0v + 1v$?
(ii) Which of VS 1 to VS 8 is used to justify the second equality: $0v + 1v = (0 + 1)v$?
(b) The second line of the proof reads:
Adding $-v$ to both sides shows that $0v = O$.
Let us understand this line in more detail. Adding $-v$ to both sides gives:
$(0v + v) + (-v) = v + (-v)$.
(i) Explain why the left hand side $(0v + v) + (-v)$ is equal to $0v$. (You need to use
three of VS 1 to VS 8.)
(ii) Explain why the right hand side $v + (-v)$ is equal to O.
