Section 2.10 Extending Linearly Independent Sets
Checkpoint 2.10.1.
Describe how to show a set \(\mathbb{S}\) spans a linear space \(\mathbb{L}\text{.}\) Describe 3 ways to show a set \(\mathbb{S}\) is linearly independent.
Checkpoint 2.10.2.
Show \(\{ \oq{1}{1}{0}{0},\oq{0}{1}{1}{0},\oq{0}{0}{1}{1} \}\) is linear independent in \(\mathbb{R}^4\text{.}\) Does it also span \(\mathbb{R}^4\text{?}\)
Checkpoint 2.10.3.
Given that \(\mathbb{S} = \{ \ot{1}{0}{0},\ot{1}{1}{0} \}\) is linearly independent in \(\mathbb{R}^3\text{,}\) find a point \(\mathbf{P}\) such that \(\mathbb{S} \cup \{\mathbf{P}\}\) is also linearly independent. Describe all possible such points.
Theorem 2.10.4.
If \(\mathbb{S}\) is a linearly independent subset of \(\mathbb{L}\) and \(\mathbf{P}\) is a point of \(\mathbb{L}\text{,}\) not in \(span\mathbb{S}\text{,}\) then \(\mathbb{S} \cup \{ \mathbf{P} \}\) is also linearly independent.
Definition 2.10.5.
A finite subset \(\mathbb{B}\) of a linear space \(\mathbb{L}\) is a basis if each point in \(\mathbb{L}\) can be written in one and only one way as a linear combination of elements of \(\mathbb{B}\text{.}\) In other words, \(\mathbb{B}\) is a basis for \(\mathbb{L}\) if it spans \(\mathbb{L}\) and is linearly independent.
Discussion Question 2.10.6.
For infinite subsets \(\mathbb{B}\) of \(\mathbb{L}\) define \(span\mathbb{B}\) to be the set of all linear combinations involving a finite number of element of \(\mathbb{B}\text{.}\) With this addition the definitions of span and linearly independent can be extended to infinite subsets of \(\mathbb{L}\text{.}\) Thus a basis may contain an infinite number of elements; however, only a finite number of them may be used when making linear combinations. What problems could occur if infinite sums were allowed for linear combinations?