A vector space has many subsets, but some are “smaller vector spaces” called subspaces. More precisely, if \(V\) is a real vector space and \(W \subseteq V\), then it is said that \(W\) is a subspace of \(V\) if \(W\) is itself a real vector space. In the post on linear maps in vector spaces, we introduced subspaces and discussed them briefly. In this post, we will discuss subspaces of vector spaces in more detail.

### When subsets are subspaces of vector spaces

Let us recall that a nonempty subset \(W\) of a real vector space \(V\) is a subspace of \(V\) if and only if for all vectors \(w_1\) and \(w_2\) in \(W\) and real numbers \(r_1\) and \(r_2\), we have $$r_1 w_1 + r_2 w_2 \in W.$$ In other words, a nonempty subset of a vector space is a subspace if and only if any linear combination of the elements of the subset is in the subset.

As a rephrased version of the discussion above, we can say:

A a nonempty subset \(W\) of a real vector space \(V\) is a subspace of \(V\) if and only if the following conditions hold:

- If \(w_1\) and \(w_2\) are elements of \(W\), then their addition \(w_1 + w_2\) is an element of \(W\).
- If \(w\) is an element of \(W\) and \(r\) is a real number, then \(rw\) is an element of \(W\).

Taking our formulations above and putting them into words, we can say the following:

**Examples**. In the following, we give some examples of subspaces of real vector spaces:

- The \(XY\) plane is a subspace of the three-dimensional real vector space \(\mathbb{R}^3\).
- A line \(L\) in \(\mathbb{R}^3\) is a subspace of \(\mathbb{R}^3\) if and only if it passes through the origin (prove this!).
- The subsets \(\{\vec{0}\}\) and \(\mathbb{R}^n\) are subspaces of \(\mathbb{R}^n\).

#### Subspaces related to linear maps between vector spaces

**Exercise**. Let \(U\) and \(V\) be real vector spaces and \(f\) be a linear map from \(U\) into \(V\). Prove that if \(W\) is a subspace of \(U\), then \(f(W)\) is a subspace of \(f(U)\).

**Solution**. Since \(0\in W\), then \(f(0)\in f(W)\). Linear maps send the zero vector to zero vector., i.e., $$f(0) = 0.$$ This means that \(f(W)\) is nonempty. Now, let \(f(w_1)\) and \(f(w_1)\) be arbitrary elements of \(f(W)\). Observe that $$r_1 f(w_1) + r_2 f(w_2) = $$ $$f(r_1w_1 + r_2w_2) \in f(W),$$ because \(f\) is a linear map. Therefore, \(f(W)\) which is a subset of \(f(U)\) is, in fact, a subspace of \(f(U)\). In a similar manner, one can prove that \(f(U)\) itself is a vector space.

**Exercise**. Let \(f\) be a linear function from a vector space \(U\) into a vector space \(V\). Assume that \(W\) is a subspace of \(V\). Prove that \(f^{-1}(W)\) is a subspace of \(U\).

### Subspaces spanned by finite subsets of vector spaces

Let \(\{v_1, \dots, v_p\}\) be a finite subset of a real vector space \(V\). It is a good exercise to see that the set of all vectors of the form $$r_1 v_1 + \dots + r_p v_p,$$ where \(r_i\)s are arbitrary real numbers, is a subspace of \(V\), denoted by $$ \hbox{span}\{v_1, \dots, v_p\}.$$

If \(W\) is a subspace of a vector space \(V\) such that $$ W = \hbox{span}\{v_1, \dots, v_p\}$$, then it is said that \(W\) is spanned (or generated) by \(\{v_i\}\)s.

**Exercise**. Let \(v_1\), \(v_2\), and \(v_3\) be elements of a real vector space \(V\). Prove that the sets $$\{v_1, v_2, v_3\}$$ and $$\{v_1, v_1 + v_2, v_1 + v_2 + v_3\}$$ span the same subspace of \(V\).

**Exercise**. Let \(v_1\), \(v_2\), and \(v_3\) be elements of a real vector space \(V\). Prove that the set $$\{v_1, v_2, v_3\}$$ is linearly independent if and only if the set $$\{v_1, v_1 + v_2, v_1 + v_2 + v_3\}$$ is linearly independent. For a discussion of linearly independent vectors see linear combination and independence.