Appendix A Notation
Symbol | Description | Location |
---|---|---|
\(x\in A\) | set membership | Definition 1.1.1 |
\(A\subseteq B\) | set inclusion | Definition 1.1.3 |
\(A\cup B\) | set union | Definition 1.1.8 |
\(A\cap B\) | set intersection | Definition 1.1.8 |
\(A-B\) | set difference | Definition 1.1.8 |
\(\{\ \}, \emptyset\) | the empty set | Definition 1.1.9 |
\(\mathbb{R}\) | the real numbers | Definition 1.1.9 |
\(\mathbb{C}\) | the complex numbers | Definition 1.1.9 |
\(\mathbb{Z}\) | the integers | Definition 1.1.9 |
\(\mathbb{Q}\) | the rational numbers | Definition 1.1.9 |
\(A\times B\) | Cartesian product | Definition 1.1.10 |
\(f\colon A\rightarrow B\) | a function from \(A\) to \(B\) | Definition 1.1.13 |
\(f(A)\) | image of the set \(A\) under \(f\) | Definition 1.1.18 |
\(\operatorname{im} f\) | image of a function \(f\) | Definition 1.1.18 |
\(f\circ g\) | the composition of \(f\) and \(g\) | Definition 1.1.21 |
\(\begin{amatrix}[c|c]A\amp \mathbb{b}\end{amatrix}\) | augmented matrix | Definition 2.2.1 |
\(A\xrightarrow{c\,r_i} B\) | scalar multiplication | Remark 2.2.6 |
\(A\xrightarrow{r_i\leftrightarrow r_j} B\) | row swap | Remark 2.2.6 |
\(A\xrightarrow{r_i+c\,r_j} B\) | replace \(r_i\) with \(r_i+c\,r_j\) | Remark 2.2.6 |
\([a_{ij}]_{m\times n}\) | 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 3.1.7 |
\(-A\) | Additive inverse of \(A\) | Definition 3.2.2 |
\(I\) | inverse matrix | Definition 3.2.3 |
\(A^{-1}\) | inverse of \(A\) | Definition 3.3.1 |
\(A^r\) | matrix power | Definition 3.3.9 |
\(f(A)\) | matrix polynomial | Definition 3.3.10 |
\(\underset{cr_i}{E}\) | 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 3.5.14 |
\(M_{mn}\) | vector space of \(m\times n\) matrices | Definition 4.1.3 |
\(\R^n\) | vector space of \(n\)-tuples | Definition 4.1.4 |
\(\{\boldzero\}\) | the zero vector space | Definition 4.1.6 |
\(\R^\infty\) | the vector space of infinite real sequences | Definition 4.1.7 |
\(F(X,\R)\) | vector space of functions from \(X\) to \(\R\) | Definition 4.1.8 |
\(\R_{>0}\) | vector space of positive real numbers | Definition 4.1.10 |
\(T_A\) | the matrix transformation associated to \(A\) | Definition 4.2.8 |
\(\rho_\alpha\) | rotation by \(\alpha\) in the plane | Definition 4.2.12 |
\(\Span S\) | the span of \(S\) | Definition 4.4.1 |
\(\val{X}\) | the cardinality of the set \(X\) | Definition 4.5.9 |
\(\dim V\) | dimension of \(V\) | Definition 4.5.12 |
\(\rank T\) | the rank of \(T\) | Definition 4.6.1 |
\(\nullity T\) | the nullity of \(T\) | Definition 4.6.1 |
\(\NS A\) | the null space of matrix \(A\) | Definition 4.6.5 |
\(\RS A\) | the row space of a matrix \(A\) | Definition 4.6.5 |
\(\CS A\) | the column space of a matrix \(A\) | Definition 4.6.5 |
\(\rank A\) | the rank of a matrix \(A\) | Definition 4.6.5 |
\(\nullity A\) | the nullity of a matrix \(A\) | Definition 4.6.5 |
\(\norm{\boldv}\) | norm of \(\boldv\) | Definition 5.1.13 |
\(d(\boldv, \boldw)\) | the distance between \(\boldv\) and \(\boldw\) | Definition 5.1.15 |
\(W^\perp\) | the orthogonal complement of \(W\) | Definition 5.2.9 |
\([P]_B^{B'}\) | change of basis matrix | Definition 6.3.1 |