§ 3. Subspaces

Definition and Examples of Subspace

Definition 3.1 : Subspace

A subset \(W\) of a vector space \(V\) over a field \(F\) is called a subspace of \(V\) if \(W\) is a vector space over \(F\) with the operations of addition and scalar multiplication defined on \(V\).

In any vector space \(V\), note that \(V\) and \(\{0\}\) are subspaces. The latter is called the zero subspace of \(V\).

Theorem 3.2 : Method for determining whether or not a given subset of a vector space is a subspace

Let \(V\) be a vector space and \(W\) a subset of \(V\). Then \(W\) is a subspace of \(V\) if and only if the following three conditions hold for the operations defined in \(V\).

• (a) \(0 \in W\).
• (b) \(x+y \in W\) whenever \(x \in W\) and \(y \in W\). (\(W\) is closed under addition.)
• (c) \(c x \in W\) whenever \(c \in F\) and \(x \in W\). (\(W\) is closed under scalar multiplication.)

Proof

If \(W\) is a subspace of \(V\), then \(W\) is a vector space with the operations of addition and scalar multiplication defined on V. Hence conditions (b) and (c) hold, and there exists a vector \(0^{\prime} \in W\) such that \(x+0^{\prime}=x\) for each \(x \in W\). But also \(x+0=x\), and thus \(0^{\prime}=0\) by Theorem 2.7. So condition (a) holds.

Conversely, because properties (VS 1), (VS 2), (VS 5), (VS 6), (VS 7), and (VS 8) hold for all vectors in the vector space, these properties automatically hold for the vectors in any subset. Thus if conditions (a), (b), and (c) hold, \(W\) is a subspace of \(V\) if the additive inverse of each vector in \(W\) lies in \(W\). But if \(x \in W\), then \((-1) x \in W\) by condition (c), and \(-x=(-1) x\) by Theorem 2.10. Hence \(W\) is a subspace of \(V\).

Example 3.3 : \(\mathrm{P}_{n}(F)\) (set of all polynomials in \(\mathrm{P}(F)\) having degree less than or equal to \(n\)) is a subspace of \(\mathrm{P}(F)\).

Let \(n\) be a nonnegative integer, and let \(\mathrm{P}_{n}(F)\) consist of all polynomials in \(\mathrm{P}(F)\) having degree less than or equal to \(n\). Since the zero polynomial has degree \(-1\), it is in \(\mathrm{P}_{n}(F)\).

Moreover, the sum of two polynomials with degrees less than or equal to \(n\) is another polynomial of degree less than or equal to \(n\), and the product of a scalar and a polynomial of degree less than or equal to \(n\) is a polynomial of degree less than or equal to \(n\). So \(\mathrm{P}_{n}(F)\) is closed under addition and scalar multiplication. It therefore follows from Theorem 3.2 that \(\mathrm{P}_{n}(F)\) is a subspace of \(\mathrm{P}(F)\).

Example 3.4 : \(\mathcal{C}(\mathbb{R})\) (set of all continuous real-valued functions defined on \(\mathbb{R}\)) is a subspace of \(\mathcal{F}(\mathbb{R}, \mathbb{R})\).

Let \(\mathcal{C}(\mathbb{R})\) denote the set of all continuous real-valued functions defined on \(\mathbb{R}\). Clearly \(\mathcal{C}(\mathbb{R})\) is a subset of the vector space \(\mathcal{F}(\mathbb{R}, \mathbb{R})\).

We claim that \(\mathcal{C}(\mathbb{R})\) is a subspace of \(\mathcal{F}(\mathbb{R}, \mathbb{R})\). First note that the zero of \(\mathcal{F}(\mathbb{R}, \mathbb{R})\) is the constant function defined by \(f(t)=0\) for all \(t \in \mathbb{R}\). Since constant functions are continuous, we have \(f \in \mathcal{C}(\mathbb{R})\). Moreover, the sum of two continuous functions is continuous, and the product of a real number and a continuous function is continuous. So \(\mathcal{C}(\mathbb{R})\) is closed under addition and scalar multiplication and hence is a subspace of \(\mathcal{F}(\mathbb{R}, \mathbb{R})\) by Theorem 3.2.

Example 3.5 : Symmetric matrices, diagonal matrices, upper triangular matrices form a subspace of \(\mathrm{M}_{n \times n}(F)\).

The transpose \(A^{t}\) of an \(m \times n\) matrix \(A\) is the \(n \times m\) matrix obtained from \(A\) by interchanging the rows with the columns: that is, \(\left(A^{t}\right)_{i j}=A_{j i}\).

A symmetric matrix is a matrix \(A\) such that \(A^{t}=A\). Clearly, a symmetric matrix must be square. The set \(W\) of all symmetric matrices in \(\mathrm{M}_{n \times n}(F)\) is a subspace of \(\mathrm{M}_{n \times n}(F)\) since the conditions of Theorem 3.2 hold:

1. The zero matrix is equal to its transpose and hence belongs to \(W\).
2. If \(A \in W\) and \(B \in W\), then \(A^{t}=A\) and \(B^{t}=B\). Thus \((A+B)^{t}=A^{t}+B^{t}=A+B\), so that \(A+B \in W\).
3. If \(A \in W\), then \(A^{t}=A\). So for any \(a \in F\), we have \((a A)^{t}=a A^{t}=a A\). Thus \(a A \in W\).

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An \(n \times n\) matrix \(M\) is called a diagonal matrix if \(M_{i j}=0\) whenever \(i \neq j\), that is, if all its nondiagonal entries are zero. Clearly the zero matrix is a diagonal matrix because all of its entries are \(0\).

Moreover, if \(A\) and \(B\) are diagonal \(n \times n\) matrices then whenever \(i \neq j\),

\[ (A+B)_{i j}=A_{i j}+B_{i j}=0+0=0 \quad \text { and } \quad(c A)_{i j}=c A_{i j}=c 0=0 \]

for any scalar \(c\). Hence \(A+B\) and \(c A\) are diagonal matrices for any scalar c. Therefore the set of diagonal matrices is a subspace of \(\mathrm{M}_{n \times n}(F)\) by Theorem 3.2.

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An \(m \times n\) matrix \(A\) is called upper triangular if all entries lying below the diagonal entries are zero, that is, if \(A_{i j}=0\) whenever \(i>j\). It can be similarly shown that the upper triangular matrices form a subspace of \(\mathrm{M}_{m \times n}(F)\).

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The trace of an \(n \times n\) matrix \(M\), denoted \(\operatorname{tr}(M)\), is the sum of the diagonal entries of \(M\); that is,

\[ \operatorname{tr}(M)=M_{11}+M_{22}+\cdots+M_{n n} \]

It can be shown that the set of \(n \times n\) matrices having trace equal to zero is a subspace of \(\mathrm{M}_{n \times n}(F)\).

Forming New Subspaces from Other Subspaces

Theorem 3.6 : Intersection of subspaces

Any intersection of subspaces of a vector space \(V\) is a subspace of \(V\).

Proof

Let \(\mathcal{C}\) be a collection of subspaces of \(V\), and let \(W\) denote the intersection of the subspaces in \(\mathcal{C}\). Since every subspace contains the zero vector, \(0 \in W\). Let \(a \in F\) and \(x, y \in W\). Then \(x\) and \(y\) are contained in each subspace in \(\mathcal{C}\). Because each subspace in \(\mathcal{C}\) is closed under addition and scalar multiplication, it follows that \(x+y\) and \(a x\) are contained in each subspace in \(\mathcal{C}\). Hence \(x+y\) and \(a x\) are also contained in \(W\), so that \(W\) is a subspace of \(V\) by Theorem 3.2.

Theorem 3.7 : Union of subspaces

Let \(W_{1}\) and \(W_{2}\) be subspaces of a vector space \(V\). \(W_{1} \cup W_{2}\) is a subspace of \(V\) if and only if \(W_{1} \subseteq W_{2}\) or \(W_{2} \subseteq W_{1}\).

Proof

(\(\Rightarrow\))
Assume that \(W_{1}\cup W_{2}\) is a subspace of \(V\). Suppose for contradiction that neither \(W_{1}\subseteq W_{2}\) nor \(W_{2}\subseteq W_{1}\) holds. Then there exist vectors

\[ u\in W_{1}\setminus W_{2} \quad\text{and}\quad v\in W_{2}\setminus W_{1}. \]

Since \(W_{1}\cup W_{2}\) is a subspace, it is closed under addition, so

\[ u+v\in W_{1}\cup W_{2}. \]

Hence either \(u+v\in W_{1}\) or \(u+v\in W_{2}\).

If \(u+v\in W_{1}\), then because \(u\in W_{1}\) and \(W_{1}\) is a subspace (hence closed under subtraction),

\[ v=(u+v)-u\in W_{1}, \]

contradicting \(v\notin W_{1}\).

If \(u+v\in W_{2}\), then because \(v\in W_{2}\) and \(W_{2}\) is a subspace,

\[ u=(u+v)-v\in W_{2}, \]

contradicting \(u\notin W_{2}\).

Both cases yield contradictions, so our assumption was false. Therefore,

\[ W_{1}\subseteq W_{2}\quad\text{or}\quad W_{2}\subseteq W_{1}. \]

(\(\Leftarrow\))
Assume that \(W_{1}\subseteq W_{2}\). Then

\[ W_{1}\cup W_{2}=W_{2}, \]

which is a subspace of \(V\). Similarly, if \(W_{2}\subseteq W_{1}\), then \(W_{1}\cup W_{2}=W_{1}\), also a subspace.

Hence \(W_{1}\cup W_{2}\) is a subspace of \(V\) if and only if \(W_{1}\subseteq W_{2}\) or \(W_{2}\subseteq W_{1}\)

The union of subspaces of \(V\) failed to be a subspace because in general, it need not be closed under addition. The sum of subspaces is a natural way to combine two subspaces \(W_{1}\) and \(W_{2}\) to obtain a subspace that contains both \(W_{1}\) and \(W_{2}\).

Definition 3.8 : Sum of Sets

If \(S_{1}\) and \(S_{2}\) are nonempty subsets of a vector space \(V\), then the sum of \(S_{1}\) and \(S_{2}\), denoted \(S_{1}+S_{2}\), is the set \(\left\{x+y: x \in S_{1}, y \in S_{2}\right\}\).

Definition 3.9 : Direct Sum

A vector space \(V\) is called the direct sum of \(W_{1}\) and \(W_{2}\) if \(W_{1}\) and \(W_{2}\) are subspaces of \(V\) such that \(W_{1} \cap W_{2}=\{0\}\) and \(W_{1}+W_{2}=V\). We denote that \(V\) is the direct sum of \(W_{1}\) and \(W_{2}\) by writing \(V=W_{1} \oplus W_{2}\).

Theorem 3.10 : \(W_1+W_2\) is the smallest subspace of \(V\) containing both \(W_1\) and \(W_2\).

Let \(W_{1}\) and \(W_{2}\) be subspaces of a vector space \(V\).

• (a) \(W_{1}+W_{2}\) is a subspace of \(V\) that contains both \(W_{1}\) and \(W_{2}\).
• (b) Any subspace of \(V\) that contains both \(W_{1}\) and \(W_{2}\) must also contain \(W_{1}+W_{2}\).

Combining (a) and (b), \(W_1+W_2\) is the smallest subspace of \(V\) containing both \(W_1\) and \(W_2\).

Proof

• (a)
  Since \(0\in W_1\) and \(0\in W_2\), we have \(0=0+0\in W_1+W_2\).

  If \(x=w_1+w_2\) and \(y=w_1'+w_2'\) are in \(W_1+W_2\), then

  \[ x+y=(w_1+w_1')+(w_2+w_2')\in W_1+W_2 \]

  because \(w_1+w_1'\in W_1\) and \(w_2+w_2'\in W_2\).

  For any scalar \(a\in F\),

  \[ ax=a(w_1+w_2)=aw_1+aw_2\in W_1+W_2 \]

  since \(aw_1\in W_1\) and \(aw_2\in W_2\).

  Thus \(W_1+W_2\) is a subspace of \(V\).

  Moreover, for \(w_1\in W_1\),

  \[ w_1=w_1+0\in W_1+W_2, \]

  so \(W_1\subseteq W_1+W_2\), and similarly \(W_2\subseteq W_1+W_2\).

• (b)
  Let \(U\) be any subspace of \(V\) such that \(W_1\subseteq U\) and \(W_2\subseteq U\). Take an arbitrary \(x\in W_1+W_2\). Then by definition there exist \(w_1\in W_1\) and \(w_2\in W_2\) with

  \[ x=w_1+w_2. \]

  Since \(W_1\subseteq U\) and \(W_2\subseteq U\), we have \(w_1,w_2\in U\). Because \(U\) is a subspace, it is closed under addition, so

  \[ x=w_1+w_2\in U. \]

  Thus every element of \(W_1+W_2\) lies in \(U\), i.e.,

  \[ W_1+W_2\subseteq U. \]

Exercise

Exercise 3.20

Prove that if \(W\) is a subspace of a vector space \(V\) and \(w_{1}, w_{2}, \ldots, w_{n}\) are in \(W\), then \(a_{1} w_{1}+a_{2} w_{2}+\cdots+a_{n} w_{n} \in W\) for any scalars \(a_{1}, a_{2}, \ldots, a_{n}\).

Exercise 3.22

Let \(F_{1}\) and \(F_{2}\) be fields. A function \(g \in \mathcal{F}\left(F_{1}, F_{2}\right)\) is called an even function if \(g(-t)=g(t)\) for each \(t \in F_{1}\) and is called an odd function if \(g(-t)=-g(t)\) for each \(t \in F_{1}\). Prove that the set of all even functions in \(\mathcal{F}\left(F_{1}, F_{2}\right)\) and the set of all odd functions in \(\mathcal{F}\left(F_{1}, F_{2}\right)\) are subspaces of \(\mathcal{F}\left(F_{1}, F_{2}\right)\).

Exercise 3.28

A matrix \(M\) is called skew-symmetric if \(M^{t}=-M\). Clearly, a skew-symmetric matrix is square.

Let \(F\) be a field. Prove that the set \(W_{1}\) of all skew-symmetric \(n \times n\) matrices with entries from \(F\) is a subspace of \(\mathrm{M}_{n \times n}(F)\). Now assume that \(F\) is not of characteristic 2, and let \(W_{2}\) be the subspace of \(\mathrm{M}_{n \times n}(F)\) consisting of all symmetric \(n \times n\) matrices. Prove that \(\mathrm{M}_{n \times n}(F)=W_{1} \oplus W_{2}\).

Exercise 3.30

Let \(W_{1}\) and \(W_{2}\) be subspaces of a vector space \(V\). Prove that \(V\) is the direct sum of \(W_{1}\) and \(W_{2}\) if and only if each vector in \(V\) can be uniquely written as \(x_{1}+x_{2}\), where \(x_{1} \in W_{1}\) and \(x_{2} \in W_{2}\).

Exercise 3.31

Let \(W\) be a subspace of a vector space \(V\) over a field \(F\). For any \(v \in V\) the set \(\{v\}+W=\{v+w: w \in W\}\) is called the coset of \(W\) containing \(v\). It is customary to denote this coset by \(v+W\) rather than \(\{v\}+W\).

• (a) Prove that \(v+W\) is a subspace of \(V\) if and only if \(v \in W\).
• (b) Prove that \(v_{1}+W=v_{2}+W\) if and only if \(v_{1}-v_{2} \in W\).

Addition and scalar multiplication by scalars of \(F\) can be defined in the collection \(S=\{v+W: v \in V\}\) of all cosets of \(W\) as follows:

\[ \left(v_{1}+W\right)+\left(v_{2}+W\right)=\left(v_{1}+v_{2}\right)+W \]

for all \(v_{1}, v_{2} \in V\) and

\[ a(v+W)=a v+W \]

for all \(v \in V\) and \(a \in F\).

• (c) Prove that the preceding operations are well defined; that is, show that if \(v_{1}+W=v_{1}^{\prime}+W\) and \(v_{2}+W=v_{2}^{\prime}+W\), then

  \[ \left(v_{1}+W\right)+\left(v_{2}+W\right)=\left(v_{1}^{\prime}+W\right)+\left(v_{2}^{\prime}+W\right) \]

  and

  \[ a\left(v_{1}+W\right)=a\left(v_{1}^{\prime}+W\right) \]

  for all \(a \in F\).

• (d) Prove that the set \(S\) is a vector space with the operations defined in (c). This vector space is called the quotient space of \(V\) modulo \(W\) and is denoted by \(V / W\).