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1.2.  Subspaces

Definition 1.2.1. A subset \(\mathbb W\) of a vector space \(\mathbb V\) over a field \(\mathbb F\) is called a subspace of \(\mathbb V\) if \(\mathbb W\) is a vector space over \(\mathbb F\) with the operations of addition and scalar multiplication defined on \(\mathbb V.\)

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

  1. \(\forall u,v\in \mathbb W \ (u+v\in \mathbb W).\)
  2. \(\forall c \in \mathbb F\ \forall u\in \mathbb W \ (cu\in \mathbb W).\)
  3. \(\exists u\in \mathbb W \ \forall v\in \mathbb W \ (v+u=v).\)
  4. \(\forall u\in \mathbb W \ (-u\in \mathbb W).\)


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

  1. \(\mathit 0\in \mathbb W.\)
  2. \(\forall u,v\in \mathbb W \ (u+v\in \mathbb W).\)
  3. \(\forall c \in \mathbb F\ \forall u\in \mathbb W \ (cu\in \mathbb W).\)

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

Example 1.2.4. The transpose \(A^{{\!\top}}\) of an \(m\times n\) matrix \(A\) is the \(n\times m\) matrix defined as \((A^{\!\top})_{ij}=A_{ji}.\) A symmetric matrix is a matrix \(A\) such that \(A^{\!\top}=A.\) It is clear that a symmetric matrix must be square, because when we suppose that \(A\) is a \(m\times n\) matrix where \(m<n\) without loss of generality, there is no entries in \(A^{\!\top}\) corresponding to the entries \(A_{in}.\)
  The set \(\mathbb W\) of all symmetric matrix in \({\mathbb F}^{n\times n}\) is a subspace of \({\mathbb F}^{n\times n}\) since Theorem 1.2.3 hold:

  1. Since \(O^{\!\top}=O,\) \(O\in \mathbb W.\)
  2. Let \(A,B\in \mathbb W.\) 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 \((A+B)\in \mathbb W.\)
  3. Let \(A\in \mathbb W\) and \(c\in \mathbb F.\) We have \(A_{ij}=A_{ji},\) thus \((cA)_{ij}=cA_{ij}=cA_{ji}=(cA)_{ij}.\) Therefore \(cA\in \mathbb W.\)


Example 1.2.5. Let \(n\) be a nonnegative integer, and let \({\mathbb F}_n[x]\) consist of all polynomials in \({\mathbb F}[x]\) having degree less than or equal to \(n.\)

  1. Since the zero polynomial \(\mathit 0\in {\mathbb F}[x]\) has degree \(-\infty,\) it is in \({\mathbb F}_n[x].\)
  2. Let \(P, Q \in {\mathbb F}[x].\) Since \(\text{deg}(P+Q)\leq \max{\lbrace \text{deg}(P),\text{deg}(Q)\rbrace }\leq n,\ (P+Q)\in {\mathbb F}_n[x].\)
  3. Let \(P \in {\mathbb F}[x].\) Since \(\text{deg}(cP)=\text{deg}(P)\leq n, \ cP\in {\mathbb F}_n[x].\)

It therefore follows from Theorem 1.2.3 that \({\mathbb F}_n[x]\) is a subspace of \({\mathbb F}[x].\)

Example 1.2.6. Let \(C(\mathbb R, \mathbb R)\) denote the set of all continuous real-valued functions defined on \(\mathbb R.\)

  1. Since constant functions are continuous functions, \(\mathit 0 \in C(\mathbb R, \mathbb R).\)
  2. Let \(f,g\in C(\mathbb R, \mathbb R).\) Since \(f+g\) is continuous function, \((f+g)\in C(\mathbb R, \mathbb R).\)
  3. 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 \({\mathbb F}[x].\)

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 \(\mathbb W\) denote the set of all \(n\times n\) diagonal matrices.

  1. Since \(O_{ij}=0\) for all \(i\) and \(j,\ O\in \mathbb W.\)
  2. Let \(A,B\in \mathbb W.\) If \(i\neq j\) then \((A+B)_{ij}=A_{ij}+B_{ij}=0+0=0.\) Thus \((A+B)\in \mathbb W.\)
  3. Let \(A\in \mathbb W\) and \(c\in \mathbb F.\) If \(i\neq j\) then \((cA)_{ij}=cA_{ij}=c0=0c=0.\) Thus \(cA\in \mathbb W.\)

It therefore follows from Theorem 1.2.3 that \(\mathbb W\) is a subspace of \({\mathbb F}^{n\times n}.\)

Theorem 1.2.8. Any intersection of subspaces of a vector space \(\mathbb V\) is a subspace of \(\mathbb V.\)


Exercises

An \(m\times n\) matrix \(A\) is called upper triangular matrix if \(A_{ij}=0\) whenever \(i>j,\) and called lower triangular matrix \(A_{ij}=0\) whenever \(i<j.\)


References

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