Get the answer to "Let AX=B be a system of linear equations,..."

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Algebra
Linear algebra
Forms of linear equations

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Answered question

2021-07-04

Let $A X = B$ be a system of linear equations, where A is an $m \times n m \times n$ matrix, X is an n-vector, and $B B$ is an m-vector. Assume that there is one solution $X = X 0$ . Show that every solution is of the form $X 0 + Y$ , where Y is a solution of the homogeneous system $A Y = 0$ , and conversely any vector of the form $X 0 + Y$ is a solution.

Answer & Explanation

Theodore Schwartz

Skilled2021-07-05Added 99 answers

First show that every solution is of the form $X 0 + Y$ where Y is a solution of the homogeneous system $A Y = 0$ . Let X be one solution. $A X = b$ .

Then $A X - A X 0 = b - b = 0 \to A (X - X 0) = 0.$

Conclude that $X - X 0 = Y$ , therefore $X = X 0 + Y$
Now show that any vector of the form $X 0 + Y$ is a solution. $A (X 0 + Y) = A X 0 + A Y = b + 0 = b$

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