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Euclidean vector spaces

Appendix A Notation

Symbol Description Location
\(x\in A\) set membership Definition 0.1.1
\(A\subseteq B\) set inclusion Definition 0.1.3
\(A\cup B\) set union Definition 0.1.8
\(A^c\) set complement Definition 0.1.8
\(A\cap B\) set intersection Definition 0.1.8
\(A-B\) set difference Definition 0.1.8
\(\{\ \}, \emptyset\) the empty set Definition 0.1.9
\(\mathbb{R}\) the real numbers Definition 0.1.9
\(\mathbb{Z}\) the integers Definition 0.1.9
\(\mathbb{Q}\) the rational numbers Definition 0.1.9
\(f\colon A\rightarrow B\) a function from \(A\) to \(B\) Definition 0.2.1
\(f(A)\) image of the set \(A\) under \(f\) Definition 0.2.6
\(\operatorname{im} f\) image of a function \(f\) Definition 0.2.6
\(f\circ g\) the composition of \(f\) and \(g\) Definition 0.2.9
\((a_1,a_2,\dots, a_n)\) \(I\)-tuple Definition 0.3.1
\(A_1\times A_2\times \cdots A_n\) Cartesian product Definition 0.3.4
\(\boldx=(x_i)_{i\in I}\) \(I\)-tuple Definition 0.3.5
\(\prod_{i\in I}A_i\) Cartesian product of the sets \(A_i\) Definition 0.3.6
\(\C\) the complex numbers Definition 0.6.1
\(\Re z\) real part of complex number \(z\) Definition 0.6.1
\(\Im z\) imaginary part of complex number \(z\) Definition 0.6.1
\(\C\) complex numbers Definition 0.6.1
\(\deg f\) degree of polynomial \(f\) Definition 0.7.5
\(R^n\) set of real \(n\)-tuples Definition 1.1.1
\(\R^\infty\) vector space of infinite real sequences Example 1.1.11
\(\R_{> 0}\) vector space of positive real numbers Example 1.1.12
\(\begin{amatrix}[c|c]A\amp \mathbb{b}\end{amatrix}\) augmented matrix Definition 2.2.1
\(A\xrightarrow{c\,r_i} B\) scalar multiplication Remark 2.2.9
\(A\xrightarrow{r_i\leftrightarrow r_j} B\) row swap Remark 2.2.9
\(A\xrightarrow{r_i+c\,r_j} B\) replace \(r_i\) with \(r_i+c\,r_j\) Remark 2.2.9
\(M_{mn}\) set of all \(m\times n\) matrices Definition 3.1.2
\([a_{ij}]_{m\times n}\) Matrix whose \(ij\)-th entry is \(a_{ij}\) Definition 3.1.3
\((A)_{ij}\) \(ij\)-th entry of the matrix \(A\) Definition 3.1.3
\(\boldzero_{m\times n}\) the \(m\times n\) zero matrix Definition 3.1.13
\(I\) inverse matrix Definition 3.2.4
\(A^{-1}\) inverse of \(A\) Definition 3.3.1
\(A^r\) matrix power Definition 3.3.10
\(f(A)\) matrix polynomial Definition 3.3.11
\(\underset{cr_i}{E}\) Scaling elementary matrix Definition 3.4.1
\(\underset{r_i\leftrightarrow r_j}{E}\) Row swap elementary matrix Definition 3.4.1
\(\underset{r_i+c\,r_j}{E}\) Row addition elementary matrix Definition 3.4.1
\(A_{ij}\) submatrix of \(A\) Definition 3.5.1
\(\det A\) determinant of \(A\) Definition 3.5.3
\(M_{ij}\) the \(ij\)-th minor of a matrix Definition 3.5.7
\(\adj A\) adjoint of a square matrix Definition 3.5.15
\(\tr A\) the trace of \(A\) Definition 4.1.18
\(\Span S\) the span of \(S\) Definition 4.2.1
\(\val{X}\) the cardinality of the set \(X\) Definition 4.4.1
\(\dim V\) dimension of \(V\) Definition 4.4.2
\(\NS A\) the null space of matrix \(A\) Definition 4.5.1
\(\RS A\) the row space of a matrix \(A\) Definition 4.5.1
\(\CS A\) the column space of a matrix \(A\) Definition 4.5.1
\(\rank A\) the rank of a matrix \(A\) Definition 4.5.12
\(\nullity A\) the nullity of a matrix \(A\) Definition 4.5.12
\(T_A\) the matrix transformation associated to \(A\) Definition 5.1.25
\(\rho_\alpha\) rotation by \(\alpha\) in the plane Definition 5.1.33
\(\rank T\) the rank of \(T\) Definition 5.2.8
\(\nullity T\) the nullity of \(T\) Definition 5.2.8
\(\underset{B\rightarrow B'}{P}\) change of basis matrix Definition 5.5.1
\(\norm{\boldv}\) norm of \(\boldv\) Definition 7.1.1
\(d(\boldv, \boldw)\) the distance between \(\boldv\) and \(\boldw\) Definition 7.1.6
\(W^\perp\) the orthogonal complement of \(W\) Definition 7.3.1