Section 2.3 Distributing over a Difference
Definition 2.3.1.
The set of all polynomials is \(\mathbb{P}= \{ c_0+c_1x+c_2x^2+\cdots+c_kx^k | \; c_i \in \mathbb{R}, k \in \mathbb{Z}^+ \}\text{.}\) The set of polynomials of degree less than n is \(\mathbb{P}_n = \{ c_0+c_1x+c_2x^2+\cdots+c_kx^k | \; c_i \in \mathbb{R}, k \in \mathbb{Z}^+, k\lt n \}\text{.}\) In both cases addition and scalar multiplication are done in the traditional way.
Checkpoint 2.3.2.
Is \(\mathbb{P}\) a linear space?
Checkpoint 2.3.3.
Is the set of polynomials of degree exactly 3 a linear space? Is the set of polynomials of degree less than 3 a linear space? Why is \(\mathbb{P}_n\) defined the way it is?
Checkpoint 2.3.4.
Write Theorem 2.3.5 as a sentence in English.
Theorem 2.3.5.
For any points \(\mathbf{P}\) and \(\mathbf{Q}\) in linear space \(\mathbb{L}\text{,}\) and any \(c\in\mathbb{R}\text{:}\)
Challenge 2.3.6.
Is \(\mathbf{0}\text{,}\) the additive identity element in a linear space, unique?