3.5.E: Problems on Linear Spaces (Exercises)
Prove that \(F^{n}\) in Example \((\mathrm{b})\) is a vector space, i.e., that it satisfies all laws stated in Theorem 1 in §§1-3; similarly for \(W\) in Example (d).
Verify that dot products in \(C^{n}\) obey the laws \((\mathrm{i})-\left(\mathrm{v}^{\prime}\right) .\) Which of these laws would fail if these products were defined by
\[
x \cdot y=\sum_{k=1}^{n} x_{k} y_{k} \text { instead of } x \cdot y=\sum_{k=1}^{n} x_{k} \overline{y}_{k} ?
\]
How would this affect the properties of absolute values given in \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right) ?\)
Complete the proof of formulas \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right)\) for Euclidean spaces. What change would result if property (ii) of dot products were restated as
\[
" x \cdot x \geq 0 \text { and } \overrightarrow{0} \cdot \overrightarrow{0}=0^{\prime \prime} ?
\]
Define orthogonality, parallelism and angles in a general Euclidean space following the pattern of §§1-3 (text and Problem 7 there). Show that \(u=\overrightarrow{0}\) iff \(u\) is orthogonal to all vectors of the space.
Define the basic unit vectors \(e_{k}\) in \(C^{n}\) exactly as in \(E^{n},\) and prove
Theorem 2 in §§1-3 for \(C^{n}\left(\text { replacing } E^{1} \text { by } C\right).\) Also, do Problem 5\((\mathrm{a})\) of §§1-3 for \(C^{n}\).
Define hyperplanes in \(C^{n}\) as in Definition 3 of §§4-6, and prove Theorem 1 stated there, for \(C^{n} .\) Do also Problems \(4-6\) there for \(C^{n}\) (replacing \(E^{1}\) by \(C )\) and Problem 4 there for vector spaces in general (replacing \(E^{1}\) by the scalar field \(F ) .\)
Do Problem 3 of §§4-6 for general Euclidean spaces (real or complex). Note: Do not replace \(E^{1}\) by \(C\) in the definition of a line and a line segment.
A finite set of vectors \(B=\left\{x_{1}, \ldots, x_{m}\right\}\) in a linear space \(V\) over \(F\) is said to be independent iff
\[
\left(\forall a_{1}, a_{2}, \ldots, a_{m} \in F\right) \quad\left(\sum_{i=1}^{m} a_{i} x_{i}=\overrightarrow{0} \Longrightarrow a_{1}=a_{2}=\cdots=a_{m}=0\right).
\]
Prove that if \(B\) is independent, then
(i) \(\overrightarrow{0} \notin B\);
(ii) each subset of \(B\) is independent \((\emptyset \text { counts as independent }) ;\) and
(iii) if for some scalars \(a_{i}, b_{i} \in F\),
\[
\sum_{i=1}^{m} a_{i} x_{i}=\sum_{i=1}^{m} b_{i} x_{i},
\]
then \(a_{i}=b_{i}, i=1,2, \ldots, m\).
Let \(V\) be a vector space over \(F\) and let \(A \subseteq V .\) By the span of \(A\) in \(V\), denoted \(\operatorname{span}(A),\) is meant the set of all "linear combinations" of vectors from \(A,\) i.e., all vectors of the form
\[
\sum_{i=1}^{m} a_{i} x_{i}, \quad a_{i} \in F, x_{i} \in A, m \in N.
\]
Show that \(\operatorname{span}(A)\) is itself a vector space \(V^{\prime} \subseteq V\) (a subspace of \(V )\) over the same field \(F,\) with the operations defined in \(V .\) (We say that A spans \(V^{\prime} .\) Show that in \(E^{n}\) and \(C^{n},\) the basic unit vectors span the entire space.