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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 |
| \(\begin{amatrix}[c|c]A\amp \mathbb{b}\end{amatrix}\) |
augmented matrix |
Definition 1.2.1 |
| \(A\xrightarrow{c\,r_i} B\) |
scalar multiplication |
Remark 1.2.6 |
| \(A\xrightarrow{r_i\leftrightarrow r_j} B\) |
row swap |
Remark 1.2.6 |
| \(A\xrightarrow{r_i+c\,r_j} B\) |
replace \(r_i\) with \(r_i+c\,r_j\)
|
Remark 1.2.6 |
| \([a_{ij}]_{m\times n}\) |
Matrix whose \(ij\)-th entry is \(a_{ij}\)
|
Definition 2.1.3 |
| \((A)_{ij}\) |
\(ij\)-th entry of the matrix \(A\)
|
Definition 2.1.3 |
| \(\boldzero_{m\times n}\) |
the \(m\times n\) zero matrix |
Definition 2.1.7 |
| \(\boldx\cdot\boldy\) |
dot product |
Definition 2.1.21 |
| \(-A\) |
Additive inverse of \(A\)
|
Definition 2.2.2 |
| \(I\) |
inverse matrix |
Definition 2.2.3 |
| \(A^{-1}\) |
inverse of \(A\)
|
Definition 2.3.1 |
| \(A^r\) |
matrix power |
Definition 2.3.9 |
| \(f(A)\) |
matrix polynomial |
Definition 2.3.10 |
| \(\underset{cr_i}{E}\) |
Scaling elementary matrix |
Definition 2.4.1 |
| \(\underset{r_i\leftrightarrow r_j}{E}\) |
Row swap elementary matrix |
Definition 2.4.1 |
| \(\underset{r_i+c\,r_j}{E}\) |
Row addition elementary matrix |
Definition 2.4.1 |
| \(A_{ij}\) |
submatrix of \(A\)
|
Definition 2.5.1 |
| \(\det A\) |
determinant of \(A\)
|
Definition 2.5.3 |
| \(M_{ij}\) |
the \(ij\)-th minor of a matrix |
Definition 2.5.7 |
| \(\adj A\) |
adjoint of a square matrix |
Definition 2.5.15 |
| \(M_{mn}\) |
vector space of \(m\times n\) matrices |
Definition 3.1.3 |
| \(\R^n\) |
vector space of \(n\)-tuples |
Definition 3.1.4 |
| \(\{\boldzero\}\) |
the zero vector space |
Definition 3.1.6 |
| \(\R^\infty\) |
the vector space of infinite real sequences |
Definition 3.1.7 |
| \(F(X,\R)\) |
vector space of functions from \(X\) to \(\R\)
|
Definition 3.1.8 |
| \(\R_{>0}\) |
vector space of positive real numbers |
Definition 3.1.10 |
| \(T_A\) |
the matrix transformation associated to \(A\)
|
Definition 3.2.8 |
| \(\rho_\alpha\) |
rotation by \(\alpha\) in the plane |
Definition 3.2.12 |
| \(\tr A\) |
the trace of \(A\)
|
Definition 3.3.15 |
| \(\NS A\) |
the null space of \(A\)
|
Definition 3.4.5 |
| \(\Span S\) |
the span of \(S\)
|
Definition 3.5.1 |
| \(\val{X}\) |
the cardinality of the set \(X\)
|
Definition 3.7.1 |
| \(\dim V\) |
dimension of \(V\)
|
Definition 3.7.4 |
| \(\rank T\) |
the rank of \(T\)
|
Definition 3.8.1 |
| \(\nullity T\) |
the nullity of \(T\)
|
Definition 3.8.1 |
| \(\NS A\) |
the null space of matrix \(A\)
|
Definition 3.8.5 |
| \(\RS A\) |
the row space of a matrix \(A\)
|
Definition 3.8.5 |
| \(\CS A\) |
the column space of a matrix \(A\)
|
Definition 3.8.5 |
| \(\rank A\) |
the rank of a matrix \(A\)
|
Definition 3.8.5 |
| \(\nullity A\) |
the nullity of a matrix \(A\)
|
Definition 3.8.5 |
| \(\norm{\boldv}\) |
norm of \(\boldv\)
|
Definition 4.1.14 |
| \(d(\boldv, \boldw)\) |
the distance between \(\boldv\) and \(\boldw\)
|
Definition 4.1.20 |
| \(W^\perp\) |
the orthogonal complement of \(W\)
|
Definition 4.3.1 |
| \(\underset{B\rightarrow B'}{P}\) |
change of basis matrix |
Definition 5.3.1 |
