Linear Algebra/Comparing Set Descriptions

''This subsection is optional. Later material will not require the work here.''

Comparing Set Descriptions
A set can be described in many different ways. Here are two different descriptions of a single set:



\{\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}z\,\big|\, z\in\mathbb{R}\} \quad\text{and}\quad \{\begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}w\,\big|\, w\in\mathbb{R}\}. $$

For instance, this set contains



\begin{pmatrix} 5 \\ 10 \\ 15 \end{pmatrix} $$

(take $$z=5$$ and $$w=5/2$$) but does not contain



\begin{pmatrix} 4 \\ 8 \\ 11 \end{pmatrix} $$

(the first component gives $$z=4$$ but that clashes with the third component, similarly the first component gives $$w=4/5$$ but the third component gives something different). Here is a third description of the same set:



\{\begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix}+\begin{pmatrix} -1 \\ -2 \\ -3 \end{pmatrix}y\,\big|\, y\in\mathbb{R}\}. $$

We need to decide when two descriptions are describing the same set. More pragmatically stated, how can a person tell when an answer to a homework question describes the same set as the one described in the back of the book?

Set Equality
Sets are equal if and only if they have the same members. A common way to show that two sets, $$S_1$$ and $$S_2$$, are equal is to show mutual inclusion: any member of $$S_1$$ is also in $$S_2$$, and any member of $$S_2$$ is also in $$S_1$$.

Exercises
/Solutions/