Question

Subspaces & Linear Combination.
see attachment:

Answer 1

Suppose
\[
w= \sum_{i=1}^r c_i v_i=\sum_{i=1}^r b_i v_i \implies\\
0=\sum_{i=1}^r (c_i -b_i)v_i\\
\text{ Since we have a basis, then }
c_i = b_i \\
\text { for every } i
\]
That prove uniqueness.
Since every element is linear combination of the basis elements. We are done.

Answer 2

Let \(w=\sum b_i v_i=\sum c_iv_i\). Consider that \(0=w-w=\sum b_iv_i-\sum c_iv_i=\sum(b_i-c_i)v_i\)
Since \(v_i\) form a basis we know they are linearly independent i.e. \(b_i-c_i=0\) therefore \(b_i=c_i\). Q.E.D.

Answer 3

dang you beat me @eliassaab

Answer 4

Thanks!