Section 2.8 When a Set Contains 0
Checkpoint 2.8.1.
Negate the following:
It is raining \(\Longrightarrow\) there are clouds in the sky
For any math problem there is a solution.
For all integers, prime \(\Longrightarrow\) odd
\(x=1, \; y=2\text{,}\) and \(z=3\)
Discussion Question 2.8.2.
Do the following mean the same thing in English? What about in math?
“For any …” “For all…” “For every…”
Checkpoint 2.8.3.
Negate the definition of \(\mathbb{S}\) being linearly independent to get a definition for \(\mathbb{S}\) being linearly dependent.
Checkpoint 2.8.4.
Show how to write the point \(\ot{a}{b}{c}\) as a linear combination of the points in \(\mathbb{S} = \{ \ot{1}{2}{0}, \ot{1}{0}{0}, \ot{1}{1}{1} \}\text{.}\) Is the set \(\mathbb{S}\) linearly independent?
Theorem 2.8.5.
If \(\mathbb{S}\) is a subset of a linear space \(\mathbb{L}\) and \(\mathbf{0} \in \mathbb{S}\text{,}\) then \(\mathbb{S}\) is linearly dependent.