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Basics of Vector Spaces


Definition. A real vector space is a nonempty set \(V\) of objects, called vectors, with two operations, viz, addition and scalar multiplication, that satisfy the following properties for all vectors \(\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\) in \(V\) and all scalars (real numbers) \(c\) and \(d\).

  1. \(\overrightarrow{u}+\overrightarrow{v}\) is in \(V\).

  2. \(\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}\)

  3. \((\overrightarrow{u}+\overrightarrow{v})+\overrightarrow{w}=\overrightarrow{u}+(\overrightarrow{v}+\overrightarrow{w})\)

  4. There is a zero vector \(\overrightarrow{0}\) such that \(\overrightarrow{u}+\overrightarrow{0}=\overrightarrow{u}\).

  5. There is a vector \(-\overrightarrow{u}\) such that \(\overrightarrow{u}+(-\overrightarrow{u})=\overrightarrow{0}\).

  6. \(c\overrightarrow{u}\) is in \(V\).

  7. \(c(\overrightarrow{u}+\overrightarrow{v})=c\overrightarrow{u}+c\overrightarrow{v}\)

  8. \((c+d)\overrightarrow{u}=c\overrightarrow{u}+d\overrightarrow{u}\).

  9. \(c(d\overrightarrow{u})=(cd)\overrightarrow{u}\).

  10. \(1\overrightarrow{u}=\overrightarrow{u}\).


  1. Scalars are elements of a field such as the set of real numbers and the set of complex numbers. If scalars are complex numbers, then \(V\) is called a complex vector space.

  2. From the definition we have the following:
    1. \(0\overrightarrow{u}=\overrightarrow{0}\)

    2. \(c\overrightarrow{0}=\overrightarrow{0}\)

    3. \(-\overrightarrow{u}=(-1)\overrightarrow{u}\)

Example. The following are real vector spaces.

  1. \(V_n\), the set of all vectors (directed line segments) in \(\mathbb R^n\).

  2. \(\mathbb R^n\) and \(\mathbb C^n\).

  3. \(\mathbb R^{\infty}\), the set of all real sequences \((a_n)=(a_1,a_2,a_3,\ldots)\).

  4. \(P_n\), the set of all real polynomials of degree at most \(n\).

  5. \(F\), the set of all real-valued functions on a set \(D\).

  6. \(L(V,W)\), the set of all linear transformations \(T:V\to W\) where \(V\) and \(W\) are real vector spaces.

  7. \(M_{m,n}(\mathbb R)\), the set of all \(m\times n\) real matrices.

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