Section 2.9 Linear Dependence
Checkpoint 2.9.1.
Is the set \(\mathbb{S} = \{x^2-1,\ x+1,\ x-1 \}\) linearly independent in \(\mathbb{P}_3\text{?}\)
Checkpoint 2.9.2.
Is the set \(\mathbb{S} = \{1,\ sin^2(x),\ cos^2(x) \}\) linearly independent in C[0,1]?
Checkpoint 2.9.3.
Use Theorem 2.7.5 to write another condition for \(\mathbb{S}\) to be linearly dependent.
Theorem 2.9.4.
Let \(\mathbb{S}\) be a subset of a linear space \(\mathbb{L}\) containing more than one point. \(\mathbb{S}\) is linearly dependent if and only if some point of \(\mathbb{S}\) is a linear combination of the other points of \(\mathbb{S}\text{.}\)
Discussion Question 2.9.5.
The definition of linear independence (Definition 2.7.1), Theorem 2.7.5, and Theorem 2.9.4 give three equivalent criteria for a set to be linearly independent. Which matches the meaning of “independent” most closely? Which seems easiest to use to prove a set is linearly independent? Which makes the most sense paired with the definition of span? What do traditional Linear Algebra texts have for the definition of linearly independent?