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Section 2.2 The Difference of Q from P

To define a specific Linear Space a set of points must be given and the actions of sum and scalar product described for these points.

Does the set \(\mathbb{R}^3\) with operations as defined in Chapter 1 on Euclidean Spaces satisfy all the conditions to be a linear space?

Which conditions of linear space does the set \(\mathbb{R}^2\) satisfy when the sum and scalar product are defined in each of the following ways:

  1. \(\op{p_1}{p_2}+\op{q_1}{q_2} = \op{0}{0}\) and \(t\op{p_1}{p_2} = \op{0}{0}\)

  2. \(\op{p_1}{p_2}+\op{q_1}{q_2} = \op{p_1 \cdot q_1}{p_2 \cdot q_2}\) and \(t\op{p_1}{p_2} = \op{p_1^t}{p_2^t}\)

Definition 2.2.3.

The set of all continuous functions defined on the interval [0,1] is denoted: C[0,1]. Given functions \(f\) and \(g\text{,}\) and real number \(c\text{,}\) define the functions \(f+g\) and \(cf\) by what they do to a given real number \(x\text{,}\) in this case: \((f+g)(x):=f(x) +g(x)\) and \((cf)(x) := c \cdot f(x)\text{.}\) Recall, a function \(f\) is continuous at \(x\) if \(\lim_{w \rightarrow x} f(w) = f(x)\text{.}\)

For the graphs of continuous functions \(f\) and \(g\) given below, sketch graphs of the functions \(f+g\) and \(3g\text{.}\) Are these functions continuous on [0,1]? Will the sum of continuous functions always be continuous? Will a scalar multiple of a continuous function always be continuous? A good justification will use the definition of continuous.

\(f(x)\)

\(g(x)\)

The function f.
The function g.

Definition 2.2.6.

Let \(\mathbf{P}\) and \(\mathbf{Q}\) be points in linear space \(\mathbb{L}\text{.}\) The difference of \(\mathbf{Q}\) from \(\mathbf{P}\) is the point, written \(\mathbf{Q}-\mathbf{P}\text{,}\) that when added to \(\mathbf{P}\) results in \(\mathbf{Q}\text{.}\) In symbols: \(\mathbf{Q}-\mathbf{P}\) is the vector such that \(\mathbf{P} + (\mathbf{Q}-\mathbf{P}) = \mathbf{Q}\text{.}\)