17.1 The General Linear-Systems Problem

#Type/Definition #Topic/Linear_Algebra

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Examples: 17.4 Model of Airplane Descent Path to Landing, 17.5 Solving GLSP for Airplane Descent Path, 17.6 Toy GLSP
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Generalizations: N/A

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The General Linear-Systems Problem

Let $m, n \in N$ . Let $A \in R^{m \times n}$ be a given rectangular matrix and $\vec{b} \in R^{m}$ be a given vector. Then, the general linear-systems problem is to find an unknown vector $\vec{x} \in R^{n}$ such that

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

Remark. Unlike 12.1 The Square Linear-Systems Problem, the GLSP uses a rectangular matrix which is not allowed to be nonsingular, Additionally, there are usually many (infinite!) number of solutions.

! Sometimes, a problem may start with a square matrix but the problem itself may still be a general linear systems problem. To be a nonsingular linear systems problem, the matrix MUST be nonsingular.

\begin{aligned} NLSP or GLSP & [A]_{n \times n} [\vec{x}]_{n \times 1} & = [\vec{b}]_{n \times 1} \\ GLSP & {[\begin{array}{c} A \end{array}]}_{m \times n} {[\begin{array}{c} \vec{x} \end{array}]}_{n \times 1} & = [\vec{b}]_{m \times 1} \\ GLSP or Least Squares & {[\begin{array}{c} A \end{array}]}_{m \times n} {[\begin{array}{c} \vec{x} \end{array}]}_{n \times 1} & = {[\begin{array}{c} \vec{b} \end{array}]}_{m \times 1} \end{aligned}

However, the strategy we developed for solving the NLSP will be similar to our strategy for solving the GLSP.
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Remark. Most times when we are solving the GLSP, we are thinking about short, wide matrices that provide an infinite number of solutions.
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