Consider a solution \(\displaystyle{x}_{{1}}\) of the linear system Ax=b. Justify the facts stated in parts (a) and (b):

Hence

\(\displaystyle{A}{x}_{{h}}={0}\)

Add equation

\(\displaystyle{A}{x}_{{1}}+{a}{x}_{{h}}={b}+{0}={b}\)

\(\displaystyle{A}{\left({x}_{{1}}+{x}_{{h}}\right)}={b}\)

Hence

\(\displaystyle{\left({x}_{{1}}+{x}_{{h}}\right)}\) is asolution of \(\displaystyle{A}{x}={b}\) since \(\displaystyle{x}={x}_{{1}}+{x}_{{h}}\) satisfies this equation.

b) If \(\displaystyle{x}_{{2}}\) is another solution of the system Ax=b, then \(\displaystyle{x}_{{2}}-{x}_{{1}}\) is asolution of the system Ax=0

\(\displaystyle{A}{x}_{{2}}={b}\)

\(\displaystyle{A}{x}_{{2}}={A}{x}_{{1}}={b}-{b}={0}\)

\(\displaystyle{A}{\left({x}_{{2}}-{x}_{{1}}\right)}={0}\)

Hence \(\displaystyle{\left({x}_{{2}}-{x}_{{1}}\right)}\) is solution of

\(\displaystyle{A}{x}={0}\)

Since \(\displaystyle{x}={x}_{{2}}-{x}_{{1}}\) satisfies the equation.

Totally puzzled here