Subspaces of $\mathbb{R}^n$

Chapter 5 • Section 5-1

"Not all sets are created equal... some are spaces! 🌌"

📦 Definition of a Subspace

A subset $U$ of $\mathbb{R}^n$ is a subspace if it satisfies three conditions:

  • S1. Zero Vector: $\vec{0} \in U$.
  • S2. Closed under Addition: If $\vec{u}, \vec{v} \in U$, then $\vec{u} + \vec{v} \in U$.
  • S3. Closed under Scalar Multiplication: If $\vec{u} \in U$ and $c \in \mathbb{R}$, then $c\vec{u} \in U$.

Examples (Subspaces)

  • The set $\{\vec{0}\}$ (Zero subspace).
  • $\mathbb{R}^n$ itself.
  • Any line through the origin.
  • Any plane through the origin.

Non-Examples

  • A line NOT through the origin (fails S1).
  • The set of integers $\mathbb{Z}^n$ (fails S3).
  • The first quadrant (fails S3 with negative scalars).

🕸️ Spanning Sets

Linear Combinations

A vector $\vec{w}$ is a linear combination of vectors $\vec{v}_1, \dots, \vec{v}_k$ if there exist scalars $c_1, \dots, c_k$ such that:

$$ \vec{w} = c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_k\vec{v}_k $$

Definition: Span

The set of all possible linear combinations of a set of vectors $\{\vec{v}_1, \dots, \vec{v}_k\}$ is called the span of those vectors, denoted by $\text{span}\{\vec{v}_1, \dots, \vec{v}_k\}$.

Theorem

If $\vec{v}_1, \dots, \vec{v}_k$ are in $\mathbb{R}^n$, then $\text{span}\{\vec{v}_1, \dots, \vec{v}_k\}$ is a subspace of $\mathbb{R}^n$.

📝 Example: Subspace Test

Let $W$ be the set of all vectors in $\mathbb{R}^3$ of the form $(a, b, a+b)$. Is $W$ a subspace?

Solution

We can write any vector in $W$ as:

$$ \begin{bmatrix} a \\ b \\ a+b \end{bmatrix} = a \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} + b \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} $$

Since every vector in $W$ is a linear combination of $\vec{v}_1 = (1,0,1)$ and $\vec{v}_2 = (0,1,1)$, we have:

$$ W = \text{span}\left\{ \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\} $$

By the theorem above, since $W$ is a span of vectors, $W$ is a subspace of $\mathbb{R}^3$.

Subspace Visualizer

Visualize why a line must pass through the origin to be a subspace. Try moving the line away from the origin and see if $2\vec{v}$ stays on the line!

Y-Intercept: 0

Is a Subspace! (Passes through origin)

Null Space and Image Space

Null Space ($\nul A$)

The set of all solutions to $A\vec{x} = \vec{0}$.

$$ \nul(A) = \{ \vec{x} \in \mathbb{R}^n \mid A\vec{x} = \vec{0} \} $$

Subspace of $\mathbb{R}^n$.

Image Space ($\im A$)

Also called Column Space ($\col A$). The set of all possible outputs $A\vec{x}$.

$$ \im(A) = \{ A\vec{x} \mid \vec{x} \in \mathbb{R}^n \} $$

Subspace of $\mathbb{R}^m$.

🧠 Knowledge Check Win $15

Which of the following is NOT a subspace of $\mathbb{R}^2$?

Hint: Check if the zero vector is included.