2.2 Subspaces - ESE 2030 📏

1Reading¶

Material related to this page, as well as additional exercises, can be found in ALA Ch. 2.2 and LAA 4.1.

2Learning Objectives¶

By the end of this page, you should know:

the definition of a subspace
examples of (i) subspaces and (ii) subsets that are not subspaces

3Defining Subspaces¶

Vector spaces arising in applications typically consist of an appropriate subset of vectors selected from a larger vector space. These vector spaces living “inside” of other vector spaces are called subspaces.

One simple way to check if a subset $W \subset V$ of a vector space $V$ is a subsapce:

$W$ needs to be non-empty
$W$ should satisfy
$c \vv v + d \vv w \in W,$
(1)
for any vectors $\vv v, \vv w \in W$ and any scalar $c, d \in \mathbb{R}$ . Subspaces are said to be closed under addition and scalar multiplication because they satisfy (1). Subspaces must also contain the $\vv 0$ vector: can you understand why from (1)?

Example 1 (Subspaces in

\mathbb{R}^3

)

The trivial subspace $W = \{\vv 0\}$ that contains only the $\vv 0$ vector. We can check that $c \vv 0 + d \vv 0 = \vv 0 \in W$ for any scalar $c, d \in \mathbb{R}$ . This subspace $W$ is commonly referred to as a point at zero.
The entire space $\mathbb{R}^3$ : clearly $c \vv v + d \vv w \in \mathbb{R}^3$ for any vectors $\vv v, \vv w \in \mathbb{R}^3$ and any scalar $c, d \in \mathbb{R}$ .
The set of all vectors of the form $\bm v \\ 2v \\ 3v \em$ . We pick two vectors $\vv v = \bm v \\ 2v \\ 3v \em, \vv w = \bm w \\ 2w \\ 3w \em$ and two scalars $c d \in \mathbb{R}$ . Then,
$c \vv v + d \vv w = \bm cv+dw \\ 2cv + 2dw \\ 3cv + 3dw \em = \bm cv+dw \\ 2(cv + dw) \\ 3(cv + dw) \em = \bm x \\ 2x \\ 3x \em, \ x = cv+dw$
(2)
is also in our set. This subspace is a line parallel to $\bm 1 \\ 2 \\ 3\em$ (stretched or shrunk by $v$ ).
The set of all vectors of the form $\bm x \\ y \\ 0 \em$ . We pick two vectors $\vv v = \bm x \\ y \\ 0 \em, \vv w = \bm \hat{x} \\ \hat{y} \\ 0 \em$ and two scalars $c d \in \mathbb{R}$ . Then,
$c \vv v + d \vv w = \bm cx+d\hat{x} \\ cy + d\hat{y} \\ 0 \em = \bm \tilde{x} \\ \tilde{y} \\ 0 \em, \ \tilde{x} = cx+d\hat{x}, \tilde{y} = cy + d\hat{y}$
(3)
is also in our set. This subspace the xy-plane.

In general, subspaces of $\mathbb{R}^3$ are one of the above four types: a point, a line, a plane, or all of $\mathbb{R}^3$ .

Example 3 (Subspaces of Discrete Time Signals)

Here, our base vector space is $\mathbb S$ , the space of all doubly infinite signals $\{y_k\} = (..., y_{-2}, y_{-1}, y_0, y_1, y_2 ...)$ .

The set $\mathbb{S}_{0:T}$ that are zero for all indices except $k = \{ 0, 1, 2, ..., T\}$ : if $\{ y_k \}, \{z_k\} \in \mathbb S_{0:T}$ then $c\{y_k\} + d\{z_k\} = \{cy_k + dz_k\}$ is also zero for all indices not in $\{0, 1, ..., T\}$ for any $c, d, \in \mathbb R$ .
The set $\Sigma_0$ of signals that sum to zero and are absolutely convergent, i.e., $\{y_k\} \in \Sigma_0$ if and only if

\begin{align*} \sum_{k = -\infty}^{\infty}{y_k} = 0, \quad \sum_{k = -\infty}^{\infty}|y_k| < \infty \end{align*}

(4)

To check this, we compute the sum of $c\{y_k\} + d\{z_k\} = \{cy_k + dz_k\}$ :

\begin{align*} \sum_{k = -\infty}^{\infty}|cy_k + dz_k| &\leq |c|\sum_{k = -\infty}^{\infty}|y_k| + |d|\sum_{k = -\infty}^{\infty}|z_k| < \infty \end{align*}

(5)

where we have used that $|cy_k + dz_k|\leq |cy_k| + |dz_k| = |c||y_k| + |d||z_k|$ , and

\begin{align*} \sum_{k = -\infty}^{\infty}{cy_k + dz_k} &= c\sum_{k = -\infty}^{\infty}{y_k} + d\sum_{k = -\infty}^{\infty}{z_k} \\ &= c\cdot 0 + d\cdot 0 \quad \text{since $\sum y_k = \sum z_k = 0 $}\\ &= 0 \end{align*}

(6)

Note that we need the absolute convergence condition, i.e., that $\sum_{k = -\infty}^{\infty}|y_k| < \infty$ to split up the infinite sum above into two sums (check back on your Math 1400/1410 notes to see why!).