Subspaces

Subspaces: Definition

A subset \(U\) of a vector space \(V\) is called a subspace of \(V\) if \(U\) is itself a vector space using the same addition and scalar multiplication as \(V\).

In practice, we do not need to check all the vector space axioms again. From Axler:

Conditions for a Subspace

Let \(V\) be a vector space over a field \(\mathbb{F}\), and let \(U \subseteq V\). Then \(U\) is a subspace of \(V\) if and only if the following three conditions hold:

\[\begin{aligned} &\text{additive identity:} && 0 \in U, \\ &\text{closed under addition:} && u, w \in U \implies u + w \in U, \\ &\text{closed under scalar multiplication:} && a \in \mathbb{F} \text{ and } u \in U \implies au \in U. \end{aligned}\]

Proof

Think about why we don’t need to show that in \(U\) other properties of vector spaces also hold.

Examples of Subspaces (from Axler)

1. A subset of \(\mathbb{F}^4\)

If \(b \in \mathbb{F}\), then

\[\{(x_1, x_2, x_3, x_4) \in \mathbb{F}^4 : x_3 = 5x_4 + b\}\]

is a subspace of \(\mathbb{F}^4\) if and only if \(b = 0\).

Why: the zero vector \((0,0,0,0)\) can be in this set only if

\[0 = 5(0) + b,\]

which means \(b = 0\).

2. Continuous functions on \([0,1]\)

The set of continuous real-valued functions on the interval \([0,1]\) is a subspace of \(\mathbb{R}^{[0,1]}\).

Note: \(A^B\), such as \(\mathbb{R}^{[0,1]}\), means the set of functions from \(B\) to \(A\).

3. Differentiable functions on \(\mathbb{R}\)

The set of differentiable real-valued functions on \(\mathbb{R}\) is a subspace of \(\mathbb{R}^{\mathbb{R}}\).

4. Differentiable functions with \(f'(2) = b\)

The set of differentiable real-valued functions \(f\) on the interval \((0,3)\) such that

\[f'(2) = b\]

is a subspace of \(\mathbb{R}^{(0,3)}\) if and only if \(b = 0\).

Why?

Define \(S_b = \{f : (0,3) \to \mathbb{R} : f \text{ is differentiable and } f'(2) = b\}.\)

For \(S_b\) to be a subspace, it must contain the zero function. The zero function is

\[0(x) = 0.\]

Therefore,

\[0'(x) = 0,\]

so in particular

\[0'(2) = 0.\]

Thus the zero function belongs to \(S_b\) only when \(b = 0\).

5. Subspaces of \(\mathbb{R}^2\)

The subspaces of \(\mathbb{R}^2\) are exactly \(\{0\}, \quad \mathbb{R}^2,\) and all lines in \(\mathbb{R}^2\) passing through the origin.

6. Subspaces of \(\mathbb{R}^3\)

The subspaces of \(\mathbb{R}^3\) are exactly \(\{0\}, \quad \mathbb{R}^3,\) all lines in \(\mathbb{R}^3\) passing through the origin, and all planes in \(\mathbb{R}^3\) passing through the origin.