18.7 Superposition for Solution to GLSP

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

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

Superposition for Solution to

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

Suppose $A \in R^{m \times n}$ and $\vec{b} \in R^{m}$ are "given," with $\vec{b} \in Span {A (:, k)}_{k = 1}^{n}$ . Suppose that ${\vec{x}}^{*} \in R^{n}$ is a particular solution to inhomogeneous linear systems

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

Then, any solution to our linear system problem can be written as

\vec{x} = {\vec{x}}^{*} + \vec{z}

where $\vec{z}$ is any trivial solution to the 18.3 Homogenous Linear System $A \cdot \vec{x} = \vec{0}$ .

Let's take a look at a particular solution.

Example

${\vec{x}}^{*} = [\begin{matrix} 2.5 \\ 15.0 \\ 0.0 \end{matrix}]$

If we wanted to find solutions to the HLSP $A \vec{x} = 0$ , we can study the related problem

[\begin{matrix} 1 & 0 & - 1.25 \\ 0 & 1 & 3.00 \end{matrix}] [\begin{matrix} a_{0} \\ a_{1} \\ a_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]

Because our matrix is in 17.3 Reduced Row Echelon Form, we see the trivial solution

\begin{aligned} U (:, 3) & = [\begin{array}{c} - 1.25 \\ 3.00 \end{array}] \\ = - 1.25 [\begin{array}{c} 1 \\ 0 \end{array}] + 3.00 [\begin{array}{c} 0 \\ 1 \end{array}] \\ = - 1.25 \cdot U (:, 1) + 3.00 \cdot U (:, 2) \\ ⟹ 1.25 \cdot U (:, 1) + 3 \cdot U (:, 2) + 1 \cdot U (:, 3) & = 0 \\ ⟹ z_{1} & = [\begin{array}{c} 1.25 \\ 3.00 \\ 1.00 \end{array}] \end{aligned}

We can conclude that any solution to our original system should take the form

\vec{x} = {\vec{x}}^{*} + c_{1} {\vec{z}}_{1} = [\begin{matrix} 2.5 \\ 15.0 \\ 0.0 \end{matrix}] + c_{1} [\begin{matrix} 1.25 \\ - 3.00 \\ 1.00 \end{matrix}]

In other words, there is no unique interpolating quadratic function.

Remark. Any linear combination of solutions to the 18.3 Homogenous Linear System also solves this problem. If we can find a maximal set of linearly independent vectors that solves the system, we can characterize ALL solutions to the HLSP as a linear combination of these vectors.
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