Subspaces

DEFINITION 3.1.6 (Vector Subspace)   Let $S$ be a NON-EMPTY SUBSET of $V.$ $S ({\mathbb{F}})$ is said to be a subspace of $V ({\mathbb{F}})$ if $\alpha {\mathbf u}+ \beta {\mathbf v}\in S$ whenever $\alpha, \beta \in {\mathbb{F}}$ and ${\mathbf u}, {\mathbf v}\in S;$ where the vector addition and scalar multiplication are the same as that of $V ({\mathbb{F}}).$

Remark 3.1.7   Any subspace is a vector space in its own right with respect to the vector addition and scalar multiplication that is defined for $V ({\mathbb{F}}).$

EXAMPLE 3.1.8  

1. Let $V ({\mathbb{F}})$ be a vector space. Then
   1. $S = \{{\mathbf 0}\},$ the set consisting of the zero vector ${\mathbf 0},$
   2. $S = V$
   are vector subspaces of $V.$ These are called trivial subspaces.
2. Let $S = \{(x,y,z)\in {\mathbb{R}}^3 : x + y - z = 0 \}.$ Then $S$ is a subspace of ${\mathbb{R}}^3.$ ($S$ is a plane in ${\mathbb{R}}^3$ passing through the origin.)
3. Let $S = \{(x,y,z) \in {\mathbb{R}}^3 : x + y + z = 3 \}.$ Then $S$ is not a subspace of ${\mathbb{R}}^3.$ ($S$ is again a plane in ${\mathbb{R}}^3$ but it doesn't pass through the origin.)
4. Let $S = \{(x,y,z) \in {\mathbb{R}}^3: z=x \}.$ Then $S$ is a subspace of ${\mathbb{R}}^3.$
5. The vector space ${\cal P}_n({\mathbb{R}})$ is a subspace of the vector space ${\cal P}({\mathbb{R}}).$
6. Conisder the vector space $V$ given in Example 5. Let $W=\{(x,0) \; : x \in {\mathbb{R}}\}.$ Then check that
   $$(x,0) \oplus (y,0) = (x+y+1, -3) \not\in W.$$
   Hence $W$ is not a subspace of $V$ .

   But if we think of $W$ as a subset of the real vector space ${\mathbb{R}}^2$ (component wise addition and scalar multiplication), then $W$ is a subspace. Check that $W=\{(x, 3) \; : x \in {\mathbb{R}}\}$ is a subspace of $V$ ($W$ represents a line passing through the point $(-1, 3)$ , the zero vector of $V$ ).

7. Consider the set $M_2({\mathbb{C}})$ given in Example 9. Then $M_2({\mathbb{C}})$ is a real as well as a complex vector space Let $W = \left\{ \begin{bmatrix}a & b \\ c & d \end{bmatrix} \; : a = \overline{d} \right\}$ be a subset of $M_2({\mathbb{C}})$ . Then verify that $W$ is a vector subspace of the real vector space $M_2({\mathbb{C}})$ . The reason being the following: the condition $a = \overline{d}$ implies that for any scalar ${\alpha}, \; {\alpha}= \overline{{\alpha}}$ , as
   $${\alpha}a = \overline{{\alpha}d} = \overline{{\alpha}} \overline{d} = \overline{{\alpha}} a.$$

EXERCISE 3.1.9  

1. Let $V$ be the set of all real numbers. Define the addition in $V$ by $x \oplus y = x + y -2$ and the scalar multiplication by ${\alpha}\odot x = {\alpha}(x-2) + 2.$ Prove that $V$ is a real vector space with respect to the operations defined above.
2. Which of the following are correct statements?
   1. Let $S = \{(x,y,z)\in {\mathbb{R}}^3 : z = x^2 \}.$ Then $S$ is a subspace of ${\mathbb{R}}^3.$
   2. Let $V ({\mathbb{F}})$ be a vector space. Let ${\mathbf x}\in V.$ Then the set $\{{\alpha}{\mathbf x}: {\alpha}\in {\mathbb{F}}\}$ forms a vector subspace of $V.$
   3. Let $W = \{ f \in C([-1,1]) : f(1/2) = 0\}.$ Then $W$ is a subspace of the real vector space, $C([-1,1]).$
3. Which of the following are subspaces of ${\mathbb{R}}^n ({\mathbb{R}})$ ?
   1. $\{(x_1, x_2, \ldots, x_n ) : x_1 \geq 0 \}.$
   2. $\{(x_1, x_2, \ldots, x_n ) : x_1 + 2 x_2 = 4 x_3 \}.$
   3. $\{(x_1, x_2, \ldots, x_n ) : x_1 {\mbox{is rational }} \}.$
   4. $\{(x_1, x_2, \ldots, x_n ) : x_1 = x_3^2 \}.$
   5. $\{(x_1, x_2, \ldots, x_n ) : {\mbox{either }} x_1 \; {\mbox{ or }} x_2 {\mbox{ or both is}} 0 \}.$
   6. $\{(x_1, x_2, \ldots, x_n ) : \vert x_1\vert \leq 1 \}.$
4. Which of the following are subspaces of $\; i) {\mathbb{C}}^n ({\mathbb{R}}) \; \; ii) {\mathbb{C}}^n ({\mathbb{C}})$ ?
   1. $\{(z_1, z_2, \ldots, z_n) : z_1 {\mbox{is real }} \}.$
   2. $\{(z_1, z_2, \ldots, z_n) : z_1 + z_2 = {\overline{z_3}} \}.$
   3. $\{(z_1, z_2, \ldots, z_n) : \mid z_1 \mid = \mid z_2 \mid \}.$