Subspaces.

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Definition: Subspaces

A subset $W$ of a vector space $V$ over a Field $F$ is called a subspace if $W$ is a vector space over $F$ is a vector space, denoted by $W \leq V$.

Note

In any vector space $V, V$ itself and $\set{0}$ are subspaces. The latter is called the zero subspace of $V$.

Theorem 1.3: Properties of Subspaces

Additionally, a subset $W$ of a vector space $V$ is a subspace of $V$ if and only if the following three properties hold.

  1. $W$ has a zero vector.
  2. $x + y \in W$ whenever $x\in W$ and $y \in W$. ($W$ is closed under addition)
  3. $cx \in W$ whenever $c \in F$ and $x \in W$. ($W$ is closed under scalar multiplication)
Proof 
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Assume that $W$ is a subspace.

This implies from the definition of subspace that $W$ itself is a vector space over $F$. then for all three of the properties are satisfiled.


Assume that the three properties holds for $W$. We need to proof that $W$ is a subspace of $V$, i.e. itself is a vector space.

  1. $w_1 + w_2 = w_2 + w_1, \forall w_1, w_2 \in W \subseteq V$
  2. $(w_1 + w_2) + w_3 = w_1 + (w_2 + w_3）, \forall w_1, w_2, w_3 \in W \subseteq V$
  3. The third property.
  4. let $c= -1 \in F$， $\forall x \in W \subseteq V, x + (-1)x = (1-1) x = 0$
  5. $1w=w, \forall w \in W \subseteq V$
  6. $\forall a,b \in F, \forall x \in W \subseteq V, (ab)x=a(bx)$
  7. $\forall a \in F, \forall x, y\in W \subseteq V, a(x+y)=ax+ay$
  8. $\forall a,b \in F, \forall x \in W \subseteq V, (a+b)x = ax+ bx$
Example: $\R^2/\R=$ XY-plane
Example: X-axis

W=${(a,0)|a\in\R}$

  1. $(0,0)\in W$
  2. $(a,0) + (b,0) = (a+b, 0) \in W, V$
  3. $\lambda \in \R, \lambda(a, 0) = (\lambda a, 0) \in W$

$\therefore W$ is a subspace of $\R^2$

Any straight line in $\R^2$ that passes through the origin is a subspace of $\R^2$

Example

$W = \set{A\in M_{n\times n}(\R)|A^T=A}=$ Set of all symmetric matrices is a subspace.

Set of all diagonal matrices is a subspace.

Set of all invertible matrices is NOT a subspace.

Set of all upper triangular matrices is a subspace.

Set of all lower triangular matrices is a subspace.

$W = \set{A\in M_{n\times n}|A^2=A}$ is NOT a subspace.

$W=\set{A\in M_{n\times n}(R)| \tr{A}=0}$ is a subspace.

$\R^n$, hyperplanes that passes through the origin $a_1x_1+a_2x_2+…+a_nx_n=0$ are subspaces.

Example: $\R^5$

$\R^5, W =\set{(x_1, x_2,x_3,x_4,x_5)\in \R^5 | x_1-x_2+x_5=0, 2x_3+7x_4=0}$. subspace.

  1. $(0,0,0,0,0) \in W$, since
    $0- 0+0 =0$
    $2\cdot0+7\cdot0=0$

  2. $(x_1, x_2,…,x_5)\in W \implies x_1-x_2+x_5=0 (1), 2x_3+7x_4=0 (2)$
    $(y_1,y_2,…,y_5) \in W \implies y_1-y_2+y_5=0 (3), 2y_3+7y_4=0 (4)$
    $(x_1+y_1,x_2+y_2,…x_5+,y_5) \in W$
    $(1)+(3) = 0, (2)+(4) = 0$

  3. trivial

Theorem 1.4

The intersection of any number (uncountable is also included) of subspaces of $V$ is again a subspace of $V$.

Remarks

Union of TWO subspaces may not be a subspace in general.

Union is a subspace if and only if one subspace is the subset of the other.