Proof Let A be an m ×n matrix. (a) Prove that the system of...

Proof Let $A$ be an $m \times n$ matrix. (a) Prove that the system of linear equations $A x=b$ is consistent for all column vectors b if and only if the rank of $A$ is $m .$ (b) Prove that the homogeneous system of linear equations $A \mathbf{x}=\mathbf{0}$ has only the trivial solution if and only if the columns of $A$ are linearly independent.

Proof Let $A$ be an $m \times n$ matrix.
(a) Prove that the system of linear equations $A x=b$ is consistent for all column vectors b if and only if the rank of $A$ is $m .$
(b) Prove that the homogeneous system of linear equations $A \mathbf{x}=\mathbf{0}$ has only the trivial solution if and only if the columns of $A$ are linearly independent.

Key Concepts

Matrix Rank

Matrix rank is the maximum number of linearly independent rows or columns in a matrix. It plays a critical role in determining the properties of a linear system. In particular, if the rank of a matrix equals the number of its rows, then its column space spans the entire codomain, ensuring that the system A x = b is consistent for every possible b.

Consistency of Linear Systems

A system of linear equations is said to be consistent if at least one solution exists for it. This occurs when the vector b is in the column space of the matrix A. If A has full row rank, every possible vector b can be expressed as a linear combination of the columns of A, guaranteeing consistency of A x = b regardless of the choice of b.

Homogeneous Systems

A homogeneous system of linear equations is one where the right-hand side is the zero vector (A x = 0). Such systems always have at least one solution, known as the trivial solution, where all variables are zero. The uniqueness of the trivial solution is a key property that connects to the concept of linear independence among the columns of A.

Linear Independence

Vectors are linearly independent if no nontrivial linear combination of them gives the zero vector. In the context of a matrix, if the columns are linearly independent, then the equation A x = 0 has only the trivial solution. This property is crucial in determining the behavior of solutions to homogeneous systems and is a fundamental concept in linear algebra.

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