Examples

EXAMPLE 3.1.4  

1. The set ${\mathbb{R}}$ of real numbers, with the usual addition and multiplication (i.e., $\oplus \equiv +$ and $\odot \equiv \cdot$ ) forms a vector space over ${\mathbb{R}}.$
2. Consider the set ${\mathbb{R}}^2 = \{(x_1,x_2): x_1, x_2 \in {\mathbb{R}}\}.$ For $x_1, x_2, y_1, y_2 \in {\mathbb{R}}$ and ${\alpha}\in {\mathbb{R}},$ define,
   $$(x_1, x_2) \oplus (y_1, y_2) = (x_1 + y_1, x_2 + y_2) \hspace{.05... ...x{ and }} \hspace{.05in} {\alpha}\odot (x_1, x_2) = ({\alpha}x_1, {\alpha}x_2).$$
   Then ${\mathbb{R}}^2$ is a real vector space.
3. Let ${\mathbb{R}}^n= \{(a_1, a_2, \ldots, a_n) : a_i \in {\mathbb{R}}, 1 \leq i \leq n \},$ be the set of $n$ -tuples of real numbers. For ${\mathbf u}=(a_1, \ldots, a_n), \; {\mathbf v}= (b_1, \ldots, b_n)$ in $V$ and ${\alpha}\in {\mathbb{R}},$ we define
   $${\mathbf u}\oplus {\mathbf v}= (a_1 + b_1, \ldots, a_n + b_n) \; {\mbox{ and }} \; \alpha \odot{\mathbf u}= (\alpha a_1, \ldots, \alpha a_n)$$
   (called component wise or coordinate wise operations). Then $V$ is a real vector space with addition and scalar multiplication defined as above. This vector space is denoted by ${\mathbb{R}}^n,$ called the real vector space of $n$ -tuples.
4. Let $V = {\mathbb{R}}^{+}$ (the set of positive real numbers). This is NOT A VECTOR SPACE under usual operations of addition and scalar multiplication (why?). We now define a new vector addition and scalar multiplication as
   $${\mathbf v}_1 \oplus {\mathbf v}_2 = {\mathbf v}_1 \cdot {\mathbf v}_2 \; {\mbox{ and }} \; \alpha \odot {\mathbf v}= {\mathbf v}^{\alpha}$$
   for all ${\mathbf v}_1, {\mathbf v}_2, {\mathbf v}\in {\mathbb{R}}^{+}$ and $\alpha \in {\mathbb{R}}.$ Then ${\mathbb{R}}^{+}$ is a real vector space with $1$ as the additive identity.
5. Let $V = \{(x,y) \; : x, y \in {\mathbb{R}}\}.$ Define $(x_1,x_2) \oplus (y_1,y_2) = (x_1+y_1+1, x_2+y_2 -3), \;\; \alpha \odot (x_1,x_2) = (\alpha x_1+ \alpha - 1, \alpha x_2 - 3 \alpha + 3)$ for $(x_1,x_2), (y_1,y_2) \in {\mathbb{R}}^2$ and $\alpha \in {\mathbb{R}}.$ Then it can be easily verified that the vector $(-1, 3)$ is the additive identity and $V$ is indeed a real vector space.

   
   
   
   

   Recall $\sqrt{-1}$ is denoted $i.$

6. Consider the set ${\mathbb{C}}= \{x+ i y : x, y \in {\mathbb{R}}\}$ of complex numbers.
   1. For $x_1+ i y_1, x_2+ i y_2 \in {\mathbb{C}}$ and ${\alpha}\in {\mathbb{R}},$ define,
      

      $$(x_1+ i y_1 ) \oplus (x_2+ i y_2) = (x_1 + x_2) + i (y_1 + y_2) \;\; {\mbox{ and }}$$

      $${\alpha}\odot (x_1 + i y_1) = ({\alpha}x_1) + i ({\alpha}y_1).$$

      
      Then ${\mathbb{C}}$ is a real vector space.
   2. For $x_1+ i y_1, x_2+ i y_2 \in {\mathbb{C}}$ and ${\alpha}+ i \beta \in {\mathbb{C}},$ define,
      

      $$(x_1+ i y_1 ) \oplus (x_2+ i y_2) = (x_1 + x_2) + i (y_1 + y_2) \hspace{.05in} {\mbox{ and }}$$

      $$({\alpha}+ i \beta) \odot (x_1 + i y_1) = ({\alpha}x_1 - \beta y_1) + i ({\alpha}y_1 + \beta x_1).$$

      
      Then ${\mathbb{C}}$ forms a complex vector space.
7. Consider the set ${\mathbb{C}}^n = \{(z_1, z_2, \ldots, z_n) : z_i \in {\mathbb{C}} {\mbox{ for }} 1 \leq i \leq n\}.$ For $(z_1, \ldots, z_n), (w_1, \ldots, w_n) \in {\mathbb{C}}^n$ and ${\alpha}\in {\mathbb{F}},$ define,
   

   $$(z_1, \ldots, z_n) \oplus (w_1, \ldots, w_n) = (z_1 + w_1, \ldots, z_n+ w_n) \hspace{.05in} {\mbox{ and }}$$

   $${\alpha}\odot (z_1, \ldots, z_n) = ({\alpha}z_1, \ldots, {\alpha}z_n).$$

   
   
   1. If the set ${\mathbb{F}}$ is the set ${\mathbb{C}}$ of complex numbers, then ${\mathbb{C}}^n$ is a complex vector space having $n$ -tuple of complex numbers as its vectors.

   2. If the set ${\mathbb{F}}$ is the set ${\mathbb{R}}$ of real numbers, then ${\mathbb{C}}^n$ is a real vector space having $n$ -tuple of complex numbers as its vectors.
   Remark 3.1.5   In Example 7a, the scalars are Complex numbers and hence $i (1, 0) = (i, 0).$ Whereas, in Example 7b, the scalars are Real Numbers and hence WE CANNOT WRITE $i (1, 0) = (i, 0).$

8. Fix a positive integer $n$ and let $M_n({\mathbb{R}})$ denote the set of all $n \times n$ matrices with real entries. Then $M_n({\mathbb{R}})$ is a real vector space with vector addition and scalar multiplication defined by
   $$A \oplus B = [a_{ij}] \oplus [b_{ij}] = [ a_{ij} + b_{ij}], \;\hspace{1in} \alpha \odot A = \alpha \odot [a_{ij}] = [ \alpha a_{ij}].$$

9. Let $M_2({\mathbb{C}})$ denote the set of all $2 \times 2$ matrices with complex entries. Then $M_2({\mathbb{C}})$ is a real vector space as well as a complex vector sapce with vector addition and scalar multiplication defined by
   $$A \oplus B = \begin{bmatrix}a_1 & a_2 \\ a_3 & a_4 \end{bmatrix} ... ...n{bmatrix}{\alpha}a_1 & {\alpha}a_2 \\ {\alpha}a_3 & {\alpha}a_4 \end{bmatrix}.$$

10. Fix a positive integer $n.$ Consider the set, ${\cal P}_n({\mathbb{R}}),$ of all polynomials of degree $\leq n$ with coefficients from ${\mathbb{R}}$ in the indeterminate $x.$ Algebraically,
    $${\cal P}_n({\mathbb{R}})= \{ a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n : a_i \in {\mathbb{R}}, 0 \leq i \leq n \}.$$
    Let $f(x), g(x) \in {\cal P}_n({\mathbb{R}}).$ Then $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ and $g(x) = b_0 + b_1 x + b_2 x^2 + \cdots + b_n x^n$ for some $a_i, b_i \in {\mathbb{R}},$ $0 \leq i \leq n.$ It can be verified that ${\cal P}_n({\mathbb{R}})$ is a real vector space with the addition and scalar multiplication defined by:
    

    $$f(x) \oplus g(x) = (a_0 + b_0) + (a_1 + b_1) x + \cdots + (a_n + b_n) x^n, \; {\mbox{ and}}$$

    $$\alpha \odot f(x) = \alpha a_0 + \alpha a_1 x + \cdots + \alpha a_n x^n \;\; {\mbox{ for }} \alpha \in {\mathbb{R}}.$$

    
    
11. Consider the set ${\cal P}({\mathbb{R}}),$ of all polynomials with real coefficients. Let $f(x), g(x) \in {\cal P}({\mathbb{R}}).$ Observe that a polynomial of the form $a_0 + a_1 x + \cdots + a_m x^m$ can be written as $a_0 + a_1 x + \cdots + a_m x^m + 0 \cdot x^{m+1} + \cdots + 0 \cdot x^p$ for any $p > m.$ Hence, we can assume $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_p x^p$ and $g(x) = b_0 + b_1 x + b_2 x^2 + \cdots + b_p x^p$ for some $a_i, b_i \in {\mathbb{R}},$ $0 \leq i \leq p,$ for some large positive integer $p.$ We now define the vector addition and scalar multiplication as
    

    $$f(x) \oplus g(x) = (a_0 + b_0) + (a_1 + b_1) x + \cdots + (a_p + b_p) x^p, \; {\mbox{ and}}$$

    $$\alpha \odot f(x) = \alpha a_0 + \alpha a_1 x + \cdots + \alpha a_p x^p \;\; {\mbox{ for }} \alpha \in {\mathbb{R}}.$$

    
    Then ${\cal P}({\mathbb{R}})$ forms a real vector space.

    Suppose that ${\cal P}({\mathbb{C}})$ is the set of all polynomials with complex coefficients, then with respect to the operations similar to what has been defined above, the set ${\cal P}({\mathbb{C}})$ is a real vector space. If we allow the scalars to be complex numbers then ${\cal P}({\mathbb{C}})$ becomes a complex vector space.

12. Let $C([-1,1])$ be the set of all real valued continuous functions on the interval $[-1, 1].$ For $f , g \in C([-1,1])$ and $\alpha \in {\mathbb{R}},$ define
    

    $$(f \oplus g ) (x) = f(x) + g (x), \; {\mbox{ and }}$$

    $$(\alpha \odot f) (x) = \alpha f(x), \; {\mbox{for all }} \; x \in [-1,1].$$

    
    Then $C([-1,1])$ forms a real vector space. The operations defined above are called POINT WISE ADDITION AND SCALAR MULTIPLICATION.

13. Let $V$ and $W$ be real vector spaces with binary operations $(+, \bullet)$ and $(\oplus, \odot),$ respectively. Consider the following operations on the set $V \times W:$ for $({\mathbf x}_1, {\mathbf y}_1), ({\mathbf x}_2, {\mathbf y}_2) \in V \times W$ and ${\alpha}\in {\mathbb{R}},$ define
    

    $$({\mathbf x}_1, {\mathbf y}_1) \oplus^{\prime} ({\mathbf x}_2, {\mathbf y}_2) = ({\mathbf x}_1 + {\mathbf x}_2, {\mathbf y}_1 \oplus {\mathbf y}_2), \; {\mbox{ and}}$$

    $${\alpha}\circ ({\mathbf x}_1, {\mathbf y}_1) = ({\alpha}\bullet {\mathbf x}_1, {\alpha}\odot {\mathbf y}_1).$$

    
    On the right hand side, we write ${\mathbf x}_1 + {\mathbf x}_2$ to mean the addition in $V,$ while ${\mathbf y}_1 \oplus {\mathbf y}_2$ is the addition in $W.$ Similarly, ${\alpha}\bullet {\mathbf x}_1$ and ${\alpha}\odot {\mathbf y}_1$ come from scalar multiplication in $V$ and $W,$ respectively. With the above definitions, $V \times W$ also forms a real vector space.

The readers are advised to justify the statements made in the above examples.

From now on, we will use ` ${\mathbf u}+{\mathbf v}$ ' in place of ` ${\mathbf u}\oplus {\mathbf v}$ ' and ` $\alpha \cdot {\mathbf u}$ or $\alpha {\mathbf u}$ ' in place of ` $\alpha \odot {\mathbf u}$ '.