Question

Consider a solution x_1 of the linear system Ax=b. Justify the facts stated in parts (a) and (b): a) If x_h is a solution of the system Ax=0, then x_1+x_h is a solution of the system Ax=b. b) If x_2 is another solution of the system Ax=b, then x_2-x_1 is a solution of the system Ax=0

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asked 2021-03-20
Consider a solution \(\displaystyle{x}_{{1}}\) of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If \(\displaystyle{x}_{{h}}\) is a solution of the system Ax=0, then \(\displaystyle{x}_{{1}}+{x}_{{h}}\) is a solution of the system Ax=b.
b) If \(\displaystyle{x}_{{2}}\) is another solution of the system Ax=b, then \(\displaystyle{x}_{{2}}-{x}_{{1}}\) is a solution of the system Ax=0

Answers (1)

2021-03-22
Question:
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.
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