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Mathematics LibreTexts

5.3: Review Problems

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    1.    Check that \(\left\{\begin{pmatrix}x\\y\end{pmatrix} \middle|\,  x,y \in \Re \right\} = \Re^{2}\) (with the usual addition and scalar multiplication) satisfies all of the parts in the definition of a vector space.



    a)    Check that the complex numbers \(\mathbb{C}= \left\{x+iy \mid i^{2}=-1, x,y\in \Re \right\}\), satisfy all of the parts in the definition of a vector space over  \(\mathbb{C}\). Make sure you state carefully what your rules for vector addition and scalar multiplication are.

    b)    What would happen if you used \(\mathbb{R}\) as the base field (try comparing to problem 1).



    a)    Consider the set of convergent sequences, with the same addition and scalar multiplication that we defined for the space of sequences: 
    \[V = \left\{f \mid f \colon \mathbb{N} \rightarrow \Re, \lim_{n \rightarrow \infty} f \in \Re \right\}\subset {\mathbb{R}^{\mathbb{N}}}\, .\]

    Is this still a vector space?  Explain why or why not.

    b)    Now consider the set of divergent sequences, with the same addition and scalar multiplication as before:
    \[V = \left\{f \mid f \colon \mathbb{N} \rightarrow \Re, \lim_{n \rightarrow \infty} f \text{ does not exist or is }\pm \infty \right\}\subset {\mathbb{R}}^{\mathbb{N}}\, .\]

    Is this a vector space? Explain why or why not.



    4.    Let \(V= \{\begin{pmatrix}x\\y\end{pmatrix} : x,y \in \Re \} = \Re^{2}\).  

    Propose as many rules for addition and scalar multiplication as you can that satisfy some of the vector space conditions while breaking some others.




    5.    Consider the set of \(2\times 4\) matrices:
    \[ V = \left\{ 
    a & b & c & d \\
    e & f & g & h 
    \mid a,b,c,d,e,f,g,h \in \mathbb{C} \right\}

    Propose definitions for addition and scalar multiplication in \(V\). Identify the zero vector in \(V\), and check that every matrix in \(V\) has an additive inverse.  




    6.    Let \(P_{3}^{\Re}\) be the set of polynomials with real coefficients of degree three or less.

    a)    Propose a definition of addition and scalar multiplication to make \(P_{3}^{\Re}\) a vector space.

    b)    Identify the zero vector, and find the additive inverse for the vector \(-3-2x+x^{2}\).

    c)    Show that \(P_{3}^{\Re}\) is not a vector space over \(\mathbb{C}\). Propose a small change to the definition of \(P_{3}^{\Re}\) to make it a vector space over \(\mathbb{C}\).




    7.    Let \(V=\{x\in \mathbb{R}|x>0\}=:\mathbb{R}_{+}\). For \(x,y\in V\) and \(\lambda\in \mathbb{R}\), define
    x\oplus y = xy\, ,\qquad \lambda \otimes x = x^{\lambda}\, .
    Prove that \((V,\oplus,\otimes,\mathbb{R})\) is a vector space.




    8.    The component in the \(i\)th row and \(j\)th column of a matrix can be labeled \(m^{i}_{j}\). In this sense a matrix is a function of a pair of integers. For what set \(S\) is the set of  \(2\times2\) matrices the same as the set \(\Re^{S}\)? Generalize to other size matrices. 




    9.    Show that any function in \(\Re^{\{*,\star,\# \}}\) can be written as a sum of multiples of the functions \(e_{*},e_{\star},e_{\#$}\) defined by 
       e_{*} (k)= \left\{\!\!
           ~~1\, , &  k=*\\
           ~~0\, , &  k=\star \\
          ~~ 0\, , & k=\# 
       e_{\star} (k)=  \left\{\!\!
           ~~0\, , &  k=*\\
           ~~1\, , &  k=\star \\
           ~~0\, , & k=\# 
       e_{\#} (k)=  \left\{\!\!
           ~~0\, , &  k=*\\
           ~~0\, , &  k=\star \\
           ~~1\, , & k=\# 
       \right. $$




    10.    Let \(V\) be a vector space and \(S\) any set. Show that the set of all functions mapping \(V\to S\), \(\textit{i.e.}\) \(V^{S}\), is a vector space.
    \(\textit{Hint:}\) first decide upon a rule for adding functions whose outputs are vectors.