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Mathematics LibreTexts

Appendix E: Summary of Notation Used

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    In addition to the notation for sets and functions (as reviewed in Appendix B), the notation for matrices and linear systems, and the common mathematical symbols reviewed in Appendix D, the following notation is used frequently in the study of Linear Algebra.

    Special Sets

    1. The set of positive integers is denoted by \(\mathbb{Z}_{+} = \{1, 2, 3, 4, \ldots\}\).
    2. The set of integers is denoted by \(\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\).
    3. The set of real numbers is denoted by \(\mathbb{R}\).
    4. The set of complex numbers is denoted by \(\mathbb{C} = \{ x + y i \ | \ x, y \in \mathbb{R} \}\). (\(\mathbb{F}\) is often used to denote a set that can equally well be chosen as either \(\mathbb{R}\) or \(\mathbb{C}\).)
    5. The set of polynomials of degree at most \(n\) in the variable \(z\) and with coefficients over \(\mathbb{F}\) is denoted by \(\mathbb{F}_{n}[z] = \left\{ a_{0} + a_{1}z + a_{2}z^{2} + \cdots + a_{n}z^{n} \ | \ a_{0}, a_{1}, \ldots, a_{n} \in \mathbb{F} \right\}\).
    6. The set of polynomials of all degrees in \(z\) with coefficients over \(\mathbb{F}\) is denoted by \(\mathbb{F}[z]\).
    7. The set of matrices of size \(m \times n\) over \(\mathbb{F}\) is denoted by \(\mathbb{F}^{m \times n}\).
    8. The general linear group of \(n \times n\) invertible matrices over \(\mathbb{F}\) is denoted by \(GL(n, \mathbb{F})\).
    9. The set of continuous functions with domain \(D \subset \mathbb{R}\) and codomain \(\mathbb{R}\) is denoted by \(\mathcal{C}(D)\), and the set of smooth (a.k.a. infinitely differentiable) functions with domain \(D \subset \mathbb{R}\) and codomain \(\mathbb{R}\) is denoted by \(\mathcal{C}^{\infty}(D)\).

    Complex Numbers

    Given \(z = x + y i \in \mathbb{C}\) with \(x, y \in \mathbb{R}\), and where \(i\) denotes the imaginary unit, we denote

    1. the additive inverse of \(z\) by \(\mathrm{-}z = (\mathrm{-}x) + (\mathrm{-}y) i\).
    2. the multiplicative inverse of \(z\) by \({\displaystyle z^{-1} = \left(\frac{x}{x^{2} + y^{2}}\right) + \left(\frac{-y}{x^{2} + y^{2}}\right) i}\), assuming \(z \neq 0\).
    3. the complex conjugate of \(z\) by \(\overline{z} = x + (\mathrm{-}y) i\).
    4. the real part of \(z\) by \(\RealPart(z) = x\).
    5. the imaginary part of \(z\) by  \(\ImaginaryPart(z) = y\).
    6. the modulus of \(z\) by \(|z| = \sqrt{x^{2} + y^{2}}\).
    7. the argument of \(z\) by \({\displaystyle \Argument(z) = \min_{\theta\,\geq\,0}\left\{ \,\theta \ | \ x = \cos(\theta),\, y = \sin(\theta)\right\}}\).

    Vector Spaces

    Let \(V\) be an arbitrary vector space, and let \(U_{1}\) and \(U_{2}\) be subspaces of \(V\).  Then we denote

    1. the additive identity of \(V\) by \(0\).
    2. the additive inverse of each \(v \in V\) by \(\mathrm{-}v\).
    3. the (subspace) sum of \(U_{1}\) and \(U_{2}\) by \(U_{1} + U_{2}\).
    4. the direct sum of \(U_{1}\) and \(U_{2}\) by \(U_{1} \oplus U_{2}\).
    5. the span of \(v_{1}, v_{2}, \ldots, v_{n} \in V\) by \(\Span\left(v_{1}, v_{2}, \ldots, v_{n}\right)\).
    6. the dimension of \(V\) by \(\dim(V)\), where

    \[ \dim(V) =\begin{cases} 0 & \mbox{if \(V = \{0\}\) is the zero vector space}, \\ n & \mbox{if every basis for \(V\) has \(n \in \mathbb{Z}_{+}\) elements in it}, \\ \infty & \mbox{otherwise}. \end{cases} \]

    1. the change of basis map with respect to a given basis \(B\) for \(V\) by \([\,\cdot\,]_{B} : V \to \mathbb{R}^{n}\), where \(V\) is assumed to be \(n\)-dimensional.


    Linear Maps

    Let \(U\), \(V\), and \(W\) denote vector spaces over the field \(\mathbb{F}\). Then we denote

    1. the vector space of all linear maps from \(V\) into \(W\) by \(\mathcal{L}(V, W)\) or \(\mathrm{Hom}_{\mathbb{F}}(V, W)\).
    2. the vector space of all linear operators on \(V\) by \(\mathcal{L}(V)\) or \(\mathrm{Hom}_{\mathbb{F}}(V)\).
    3. the composition (a.k.a. product) of \(S \in \mathcal{L}(U, V)\) and \(T \in \mathcal{L}(V, W)\) by \(T \circ S\) (or, equivalently, \(TS\)), where \((T \circ S)(u) = T(S(u))\) for each \(u \in U\).
    4. the null space (a.k.a. kernel) of \(T \in \mathcal{L}(V, W)\) by \(\kernel(T) = \{ v \in V \ | \ T(v)  = 0\}\).
    5. the range of \(T \in \mathcal{L}(V, W)\) by \(\range(T) = \{ w \in W \ | \ \) w = T(v)\) for some \(v \in V\}\).
    6. the eigenspace of \(T \in \mathcal{L}(V)\) associated to eigenvalue \(\lambda \in \mathbb{C}\) by \(V_{\lambda} = \kernel(T - \lambda\mathrm{id}_{V})\), where \(\mathrm{id}_{V}\) denotes the identity map on \(V\).
    7. the matrix of \(T \in \mathcal{L}(V, W)\) with respect to the basis \(B\) on \(V\) and with respect to the basis \(C\) on \(W\) by \(\mathcal{M}(T, B, C)\) (or simply as \(\mathcal{M}(T)\)).

    Inner Product Spaces

    Let \(V\) be an arbitrary inner product space, and let \(U\) be a subspace of \(V\).  Then we denote

    1. the inner product on \(V\) by \(\langle\cdot, \cdot\rangle\).
    2. the norm on \(V\) induced by \(\langle\cdot, \cdot\rangle\) as \(\|\cdot\| = \sqrt{\langle\cdot, \cdot\rangle}\).
    3. the orthogonal complement of \(U\) by \(U^{\perp} = \left\{ v \in V \ | \ \langle u, v \rangle = 0, \, \forall \, u \in V \right\}\).
    4. the orthogonal projection onto \(U\) by \(P_{U}\), which, for each \(v \in V\), is defined by \(P_{U}(v) = u\) such that \(v = u + w\) for \(u \in U\) and \(w \in U^{\perp}\).
    5. the adjoint of the operator \(T \in \mathcal{L}(V)\) by \(T^{*}\), where \(T^{*}\) satisfies \(\langle T(v), w \rangle = \langle v, T^{*}(w) \rangle\) for each \(v, w \in V\).
    6. the square root of the positive operator \(T \in \mathcal{L}(V)\) by \(\sqrt{T}\), which satisfies \(T = \sqrt{T}\sqrt{T}\).
    7. the positive part of the operator \(T \in \mathcal{L}(V)\) by \(|T| = \sqrt{T^{*}T}\).