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# 3.5.E: Problems on Linear Spaces (Exercises)

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Exercise $$\PageIndex{1}$$

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).

Exercise $$\PageIndex{2}$$

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 $$\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right) ?$$

Exercise $$\PageIndex{3}$$

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
$" x \cdot x \geq 0 \text { and } \overrightarrow{0} \cdot \overrightarrow{0}=0^{\prime \prime} ?$

Exercise $$\PageIndex{4}$$

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.

Exercise $$\PageIndex{5}$$

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 $$C^{n}$$.

Exercise $$\PageIndex{6}$$

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 $$F ) .$$

Exercise $$\PageIndex{7}$$

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.

Exercise $$\PageIndex{8}$$

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 $$a_{i}=b_{i}, i=1,2, \ldots, m$$.

Exercise $$\PageIndex{9}$$

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.