18.8 Existence of Solution to GLSP

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Existence of Solution to

A \cdot \vec{x} = \vec{b}

Suppose $A \in R^{m \times n}$ and $\vec{b} \in R^{m}$ are "given." Then, the linear system $A \cdot \vec{x} = \vec{b}$ has a solution iff

\vec{b} \in Span {A (:, k)}_{k = 1}^{n} .

Remark. In other words, a solution to 17.1 The General Linear-Systems Problem exists iff the vector $\vec{b}$ can be written as a 6.1 Linear Combination of Vectors of the columns of $A$ .
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