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 |