What is a Subspace?

A subspace of vectors in $\mathbb R^n$ is a set of vectors with two properties.

1. For any two vectors in the set, $\vec u$ and $\vec v$, the sum $\vec u + \vec v$ is also in the set.
2. For any real number $c$, the product $c \vec u$ is also in the set.

When a set of vectors in $\mathbb R^n$ satisfies these two properties, the set is a subspace.

Notice how the zero vector must be included in the set: when $c = 0$, the product $c\vec u =0$. This useful property gives us a quick way to determine whether a set of vectors could be a subset.

Let’s explore a few examples to demonstrate why subspaces can be useful.

Example: Solutions to a Homogeneous Linear System

What is the set of all possible solutions to the linear system $$A\vec x = 0, \qquad A=\begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 5 & 10 \end{pmatrix}, \qquad \vec x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

In other words: given this matrix $A$, for what vectors, $\vec x$, is there a solution to this system? To answer this question we can, for example, determine the set of solutions to $A\vec x= \vec 0$ by row reducing the augmented matrix $(A \, | \,\vec0)$.

$$\begin{pmatrix} 1 & 2 &\bigm| & 0 \\ 2 & 4 &\bigm| & 0\\ 5 & 10 &\bigm| & 0 \end{pmatrix} \sim \begin{pmatrix} 1 & 2 &\bigm| & 0 \\ 0 & 0 &\bigm| & 0\\ 0 & 0 &\bigm| & 0 \end{pmatrix}$$

Our system was reduced to the equation

$$x_1 + 2x_2 = 0 \qquad \text{or } \qquad x_1 = -2x_2$$

Any vector of this form is a solution to our system. A few examples of solutions are:

$$ \begin{pmatrix} 2 \\ -1 \end{pmatrix}  , \ \begin{pmatrix} 4 \\ -2 \end{pmatrix}  , \ \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} -2 \\ 1 \end{pmatrix}, \ldots$$

All of these vectors are solutions to our system. 

Is the  set of all solutions a subspace? We need to check whether our two conditions are satisfied. 

1. Closure under vector addition: suppose vectors $\vec u$ and $\vec v$ are solutions. Then their sum is also a solution: $$A(\vec u + \vec v) = A\vec u + A\vec v = \vec 0 + \vec 0 = \vec 0$$
2. Closure under scalar multiplication: if $\vec u$ is a solution and $c$ is a real number, then $c\vec u$ is also a solution: $A(c\, \vec u) = c\, A\vec u = c\, \vec 0 = \vec 0$. 

Both criteria are met, and the solutions to this linear system is a subspace. 

In fact, the solution set to any homogeneous system is a subspace, because the solutions will satisfy our two required properties. 

The set of solutions to our system is sketched below. Note that the set of solutions is a line that passes through the origin! As noted above, subspaces must include the zero vector. 

Example: The Vectors in ℝ² Whose Entries Sum to 1

You might be wondering – what is an example of a set of vectors that is not a subspace?

Let’s take $S$ to be the set of vectors in $\mathbb R^2$, where the first entry plus the second entry is equal to 1.

Is this set of vectors a subspace?

To help develop some intuition, here are a few vectors in the set:
$$S = \begin{pmatrix}  0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \end{pmatrix} \ldots$$
Every vector in this set has the form $$\begin{pmatrix}  k \\ k \end{pmatrix}, \qquad k \in \mathbb R$$

This subset is not a subspace. Take $\vec u$ to be in $S$, and $$u = \begin{pmatrix}  k_1 \\ k_1 \end{pmatrix}$$

But $ c \vec u = \begin{pmatrix}  c k_1 \\ c k_1 \end{pmatrix}$, and the sum of the entries of $c\vec u$ is no longer equal to 1 for any $c$.