Superposition and Subspace

Some terms we will be look at today:

• Superposition Principle
• Subspace

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Subspaces

Essentially, solutions of a homogeneous system of linear equations

OR

a span of a list of vectors

lets look at an example

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Example

We are presented with the following linear system:

$$\begin{array}{rl}x-z& =0\\ y+z& =0\end{array}$$

Fairly simple, yields a general solution of $(p,-p,p)$.

Lets construct a basic solution.

$$BasicSolution\to \left(\begin{array}{c}1\\ -1\\ 1\end{array}\right)$$

Please Note

Note that the general solution $(p,-p,p)$ is the given AND That the basic solution $\left(\begin{array}{c}1\\ -1\\ 1\end{array}\right)$ is the goal

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Given a subspace, what is the shortest list of vectors that span the space?

Shrinking

Let’s review some possible solutions:

$$\left(\begin{array}{c}1\\ -1\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$$

where we find that the first two work for this problem

$$\begin{array}{rl}a+b& =1\\ -b+b& =0\\ 0& =0\end{array}\to$$

becomes

$$\begin{array}{rl}a& =\frac{1}{2}\\ b& =\frac{1}{2}\\ 0& =0\end{array}$$

and finally

$$\begin{array}{rl}\vec{i}& =\frac{1}{2}\vec{g}+\frac{1}{2}\vec{h}\\ \vec{j}& =0\vec{g}+0\vec{h}\end{array}$$