Subspaces

Definition 1.2.1. 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 $$V.$$

Remark 1.2.2. Let $$V$$ be a vector space over $$F,$$ and $$W$$ a subset of $$V.$$ Because (VS1), (VS2), (VS5), (VS6), (VS7), and (VS8) hold for all vectors in $$V,$$ these automatically hold for the vectors in $$W.$$ Thus $$W$$ is a subspace of $$V$$ if and only if the following four properties are true in $$W.$$

$\forall u,v\in W \ (u+v\in W).$
$\forall c \in F\ \forall u\in W \ (cu\in W).$
$\exists u\in W \ \forall v\in W \ (v+u=v).$
$\forall u\in W \ (-u\in W).$

Theorem 1.2.3. Let $$V$$ be a vector space over $$F,$$ and $$W$$ a subset of $$V.$$ Then $$W$$ is a subspace of $$V$$ if and only if the following three properties are true in $$W.$$

$0_V\in W.$
$\forall u,v\in W \ (u+v\in W).$
$\forall c \in F\ \forall u\in W \ (cu\in W).$

Proof. Suppose that $$W$$ is a subspace of $$V.$$ By Definition 1.2.1, (2) and (3) hold, and there exists $u\in W$ such that $\forall v\in W \ (v+u=v).$ But in $$V,$$ we have $$v+0_V= v,$$ and thus, by Theorem 1.1.8, $$u=0_V.$$ Therefore (1) holds. Conversely, suppose that (1), (2), and (3) hold. Then $(-1)u\in W.$ We also have $$0u=0_V,$$ because $$u+0u=1u+0u=(1+0)u=1u=u.$$ Thus $$u+(-1)u=1u+(-1)u=(1+(-1))u=0u=0_V.$$ Therefore $-u=(-1)u\in W.$ From Remark 1.2.2, we can see that $$W$$ is a subspace of $$V.$$ $\square$

Example 1.2.4. The transpose $A^{{\!\top}}$ of an $m\times n$ matrix $$A$$ is the $n\times m$ matrix defined by $(A^{\!\top})_{ij}=A_{ji}.$ A symmetric matrix is a matrix $$A$$ such that $A^{\!\top}=A.$ It is clear that a symmetric matrix must be square. It is because that $$A$$ is a $m\times n$ matrix with $$m<n$$ implies that there are no entries in $A^{\!\top}$ corresponding to the entries $A_{in}.$ The set $$W$$ of all symmetric matrix in ${F}^{n\times n}$ is a subspace of ${F}^{n\times n}$ since Theorem 1.2.3 hold: Since $O^{\!\top}=O,$ $O\in W.$ Let $A,B\in W.$ We have $A_{ij}=A_{ji}$ and $B_{ij}=B_{ji},$ thus $(A+B)_{ij}=A_{ij}+B_{ij}=A_{ji}+B_{ji}=(A+B)_{ji}.$ Therefore $(A+B)\in W.$ Let $A\in W$ and $c\in F.$ We have $A_{ij}=A_{ji},$ thus $(cA)_{ij}=cA_{ij}=cA_{ji}=(cA)_{ij}.$ Therefore $cA\in W.$

Example 1.2.5. Let $$n$$ be a nonnegative integer, and let ${F}_n[x]$ consist of all polynomials in $${F}[x]$$ having degree less than or equal to $$n.$$ Since the zero polynomial $\mathbf 0\in {F}[x]$ has degree $-\infty,$ it is in ${F}_n[x].$ Let $P, Q \in {F}[x].$ Since $\text{deg}(P+Q)\leq \max{\lbrace \text{deg}(P),\text{deg}(Q)\rbrace }\leq n,\ (P+Q)\in {F}_n[x].$ Let $P \in {F}[x].$ Since $\text{deg}(cP)=\text{deg}(P)\leq n, \ cP\in {F}_n[x].$ It therefore follows from Theorem 1.2.3 that ${F}_n[x]$ is a subspace of $${F}[x].$$

Example 1.2.6. Let $C(\mathbb R, \mathbb R)$ denote the set of all continuous real-valued functions defined on $\mathbb R.$ Since constant functions are continuous functions, $\mathbf {0} \in C(\mathbb R, \mathbb R).$ Let $f,g\in C(\mathbb R, \mathbb R).$ Since $$f+g$$ is continuous function, $(f+g)\in C(\mathbb R, \mathbb R).$ Let $f\in C(\mathbb R, \mathbb R)$ and $c\in \mathbb R.$ Since $$cf$$ is a continuous function, $cf\in C(\mathbb R, \mathbb R).$ It therefore follows from Theorem 1.2.3 that $C(\mathbb R, \mathbb R)$ is a subspace of $${F}[x].$$

Example 1.2.7. An $n\times n$ matrix $$M$$ is called diagonal matrix if $M_{ij}=0$ whenever $i\neq j,$ that is, if all its non-diagonal entries are zero. Let $$W$$ denote the set of all $n\times n$ diagonal matrices. Since $O_{ij}=0$ for all $$i$$ and $j,\ O\in W.$ Let $A,B\in W.$ If $i\neq j$ then $(A+B)_{ij}=A_{ij}+B_{ij}=0+0=0.$ Thus $(A+B)\in W.$ Let $A\in W$ and $c\in F.$ If $i\neq j$ then $(cA)_{ij}=cA_{ij}=c0=0c=0.$ Thus $cA\in W.$ It therefore follows from Theorem 1.2.3 that $$W$$ is a subspace of ${F}^{n\times n}.$

Theorem 1.2.8. Let $$V$$ be a vector space over a field $$F$.$ Then any intersection of subspaces of $$V$$ is a subspace of $$V.$$

Proof. Let $\mathcal C$ be a set whose elements are subspaces of $$V,$$ and let $$W$$ denote the intersection of $\mathcal C.$ Since every subspace contains the zero vector, $0_V\in W.$ Let $a\in F$ and $u,v\in W.$ Then $$u$$ and $$v$$ are elements of each subspace in $\mathcal C.$ Since each subspace in $\mathcal C$ is closed under the addition and scalar multiplication, $$u+v$$ and $$cu$$ are elements of each subspace in $\mathcal C.$ Hence $$u+v$$ and $$cu$$ are also elements of $$W,$$ so that $$W$$ is a subspace of $$V$$ by Theorem 1.2.3. $\square$

Exercises

Exercise 1. Prove that

\[(aA+bB)^{\!\top} = aA^{\!\top} + bB^{\!\top}\]

for any $A,B\in F^{m\times n}$ and any $a,b\in F.$

Proof. Let $1\leq i \leq m$ and $1\leq j \leq n.$ Then we have

\[\begin{aligned} \left( (aA+bB)^{\!\top}\right)_{ij} &= (aA+bB)_{ji}\\ &= (aA)_{ji} + (bB)_{ji}\\ &= aA_{ji} + bB_{ji}\\ &= a\left( A^{\!\top}\right)_{ij} + b\left( B^{\!\top}\right)_{ij}\\ &= \left( aA^{\!\top}\right)_{ij} + \left( bB^{\!\top}\right)_{ij}\\ &= \left( aA^{\!\top} + bB^{\!\top}\right)_{ij}. \end{aligned}\]

Therefore $(aA+bB)^{\!\top} = aA^{\!\top} + bB^{\!\top}.$ $\square$

Exercise 2. Prove that $A+A^{\!\top}$ is symmetric for any square matrix $$A.$$

References

Friedberg, S. H., Insel, A. J., & Spence, L. E. (2025). Linear algebra (5th ed.). Pearson Education South Asia Pte Ltd.

External

1.2. Subspaces

Exercises

References